Let \(A\) and \(B\) be two graph and \(P(A,z)\) and \(P(B,z)\) are their chromatic polynomial, respectively. The two graphs \(A\) and \(B\) are said to be chromatic equivalent denoted by \( A \sim B \) if \(P(A,z)=P(B,z)\). A graph \(A\) is said to be chromatically unique(or simply \(\chi\)- unique) if for any graph \(B\) such that \(A\sim B \), we have \(A\cong B\), that is \(A\) is isomorphic to \(B\). In this paper, the chromatic uniqueness of a new family of \(6\)-bridge graph \(\theta(r,r,s,s,t,u)\) where \(2\leq r\leq s \leq t\leq u\) is investigated.
Let \(A\) be a finite undirected graph with a vertex set \(V(A)\) and an edge set \(E(A)\). A function \(f: V(A) \rightarrow \{1, \dots, k\}\) is called a proper coloring if for any two adjacent vertices \(x\) and \(y\) (i.e., \(xy \in E(A)\)), it holds that \(f(x) \neq f(y)\). The chromatic polynomial of the graph \(A\), denoted by \(P(A, z)\), is defined as the number of all proper colorings of \(A\). Consider two graphs \(A\) and \(B\) with their respective chromatic polynomials \(P(A, z)\) and \(P(B, z)\). These graphs are said to be chromatically equivalent, denoted by \(A \sim B\), if \(P(A, z) = P(B, z)\). A graph \(A\) is termed a chromatically unique graph if no other graph shares the same chromatic polynomial as \(A\).
For each integer \(k \geq 2\), let \(\theta_{k}\) denote the multi-graph with two vertices and \(k\) edges. Any subdivision of \(\theta_{k}\) is referred to as a multi-bridge graph or a \(k\)-bridge graph, denoted by \(\theta(y_{1}, y_{2}, y_{3}, \dots, y_{k})\), where \(y_{1}, y_{2}, \dots, y_{k} \in \mathbb{N}\) and \(y_{1} \leq y_{2} \leq \dots \leq y_{k}\). The graph \(\theta(y_{1}, y_{2}, y_{3}, \dots, y_{k})\) is obtained by replacing the edges of \(\theta_{k}\) with paths of lengths \(y_{1}, y_{2}, y_{3}, \dots, y_{k}\), respectively. Consequently, the graph \(\theta(y_{1}, y_{2}, y_{3}, \dots, y_{k})\) possesses \(y_1 + y_2 + \dots + y_k – k + 2\) vertices and \(y_1 + y_2 + \dots + y_k\) edges.
The chromaticity of \(k\)-bridge graphs has been extensively studied by numerous researchers. A \(2\)-bridge graph, which is essentially a cycle graph, is known to be \(\chi\)-unique. The theta graph, a type of \(3\)-bridge graph, is denoted by \(\theta(1,y_{1},y_{2})\). Chao and Whitehead [1] established that every theta graph is \(\chi\)-unique. Extending their work, Loerinc [2] demonstrated that all \(3\)-bridge graphs are \(\chi\)-unique. The chromaticity of \(4\)-bridge graphs was successfully addressed by Chen et al. [3] and Xu et al. [4]. Research on the chromaticity of \(5\)-bridge graphs has been conducted by several scholars, as cited in [5-8].
Theorem 1. (Xu et al. [3] ) For \(k \geq 2\), the graph \(\theta_{k}(h)\) is \(\chi\) unique.
Theorem 2. (Dong et al. [9] ) If 2 \(\leq y_{1} \leq y_{2} \leq\dots\leq y_{k} < y_{1} + y_{2}\) where \(k \geq 3\), then the graph \(\theta ( y_{1},y_{2},\dots,y_{k})\) is \(\chi\)-unique.
For any graph \(A\) and real number \(z\), write \[\nonumber Q(A,z)=(-1)^{1+|E(A)|}(1-z)^{|V(A)|+|E(A)|+1}P(A,1-z).\]
Theorem 3. (Dong et al. [9] ) For any \(k\), \(y_{1},y_{2},…,y_{k} \in N\), \[Q(\theta (y_{1},y_{2},…,y_{k}),z)= z \prod^{k}_{i=1}(z^{y_{i}}-1)-\prod^{k}_{i=1}(z^{y_{i}}-z)\tag{1}\]
Theorem 4. (Dong et al. [9] ) For any graph \(A\) and B,
1. If \(B \sim A\), then Q(B,z)=
Q(A,z).
2. If Q(B,z)= Q(A,z) and v(B)= v(A), then \(B\sim A\).
Theorem 5. (Dong et al. [9] ) Suppose that \(\theta (y_{1},y_{2},\dots,y_{k}) \sim \theta (x_{1},x_{2},…,x_{k} )\), where \(k\geq 3\), \(2 \leq y_{1}\leq y_{2}\leq\dots\leq y_{k}\) and \(2 \leq x_{1}\leq x_{2}\leq\dots\leq x_{k}\), then \(y_{i} = x_{i}\) for all \(i= 1,2,3,…k.\)
Theorem 6. (Dong et al. [9] ) Let \(B \sim \theta
(y_{1},y_{2},\dots,y_{k})\) ,where \(k\geq 3\) and \(y_{i}\geq 2\) for all i, then one of them
is true:
1. \(B\) \(\cong \theta (
y_{1},y_{2},\dots,y_{k})\)
2. \(B\) \(\in g_{e}(\theta( x_{1},x_{2},\dots,x_{k})
C_{x_{i+1}},\dots, C_{x_{k+1}})\), where \(3 \leq t \leq k-1\) and \(x_{i} \geq 2\) for all \(i=1,2,3,..k.\)
Theorem 7. (Dong et al. [9] ) Let \(k, t ,x_{1},x_{2},\dots,x_{k} \in N\) where \(3\leq t\leq k-1\) and \(x_{i} \geq 2\) for all \(i= 1,2,3,\dots,k\). If \(B \in g_{e} (\theta ( x_{1},x_{2},…,x_{t}),C_{x_{t+1}+1},…,C_{x_{k+1}})\), then \[\label{equation1} Q(B,z)= z\prod^{k}_{i=1}(z^{x_{i}}-1)- \prod^{t}_{i=1}(z^{x_{i}}- z) \prod^{k}_{i=t+1}(z^{x_{i}} – 1).\tag{2}\]
Theorem 8. (Koh & Teo [])If \(A \sim
B\), then
1. \(v(A)= v(B)\),
2. \(e(A)= e(B)\),
3. \(g(A)= g(B)\),
4. \(A\) and \(B\) have the same number of shortest
cycle.
where \(v(A)\), \(v(B)\), \(e(A)\), \(e(B)\), \(g(A)\) and \(g(B)\) denote the number of vertices, the
number of edges and the girth of \(A\)
and \(B\), respectively.
The chromaticity on several families of \(6\)-bridge graph has been done by several authors which are given below.
Lemma 9. [11] A 6-bridge graph \(\theta(y_{1},y_{2},\dots,y_{6})\) is \(\chi\) unique if the positive integer \(y_{1},y_{2},\dots,y_{6}\) assume exactly two distinct values.
Lemma 10. [13] The graph 6-bridge \(\theta (3,3,3,s,s,t)\), where \(r\leq s\leq t\), is \(\chi\)-unique.
Lemma 11. [13] The graph \(6\)-bridge \(\theta (r,r,r,s,s,t)\), where \(r\leq s\leq t\), is \(\chi\)-unique.
Lemma 12. [14] The graph \(6\)-bridge \(\theta (3,3,3,s,t,u)\), where \(3\leq s\leq t\), is \(\chi\)-unique.
Lemma 13. [15] The graph \(6\)-bridge \(\theta (r,r,s,s,t,t)\), where \(r\leq s\leq t\), is \(\chi\)-unique.
Lemma 14. [16] The graph \(6\)-bridge \(\theta (r,r,s,s,s,t)\), where \(r\leq s\leq t\), is \(\chi\)-unique.
Lemma 15. [18] The graph \(6\)-bridge \(\theta (r,r,r,s,t,u)\), where \(r\leq s\leq t\leq u\), is \(\chi\)-unique.
In this paper, we have extended this study to a new family of \(6\)-bridge graph \(\theta(r,r,s,s,t,u)\) where \(2\leq r\leq s \leq t\leq u\) and showed that this family of \(6\)-bridge graph is chromatically unique.
In this section we present our main result on the chromaticity of \(6\)– bridge graph.
Theorem 16. The \(6\)-bridge graph \(\theta (r,r,s,s,t,u)\) where \(r\leq s\leq t\leq u\) is chromatically unique.
Proof. Let \(A\) be the
\(6\)-bridge graph of the form \(\theta (r,r,s,s,t,u)\) and \(2\leq r\leq s\leq t\leq u\). By Theorem 2, \(A\) is \(\chi\) unique if \(u < 2r\). Suppose \(r \geq 2\) and \(B \sim A\). We shall solve \(Q(A)= Q(B)\) to get all the solutions. The
lowest remaining power (l.r.p) means the lowest remaining power of \(z\) in the expression after simplification
and highest remaining power (h.r.p) mean the maximum power of \(z\) in the expression after simplification.
By Theorem 8, \(g(A)= g(B)=
2r\) and \(B\) has the same
number of shortest cycles as \(A\).
Thus, we have \[2r + 2s + t + u = x_{1} +
x_{2} + x_{3} + x_{4} + x_{5} + x_{6}.\tag{3}\]] By Theorem 6 and 7, there are three
cases to consider, that are
\(B \in g_{e} (\theta (x_{1},x_{2},x_{3}),
C_{x_{4}+1}, C_{x_{5}+1},C_{x_{6}+1})\), where \(2 \leq x_{1}\leq
x_{2} \leq x_{3}\) and \(2 \leq
x_{4},x_{5},x_{6}\), or
\(B \in g_{e} (\theta
(x_{1},x_{2},x_{3},x_{4}),
C_{x_{5}+1},C_{x_{6}+1})\) , where \(2
\leq x_{1}\leq x_{2} \leq
x_{3} \leq x_{4 }\) and \(2 \leq
x_{5},x_{6}\), or
\(B \in g_{e} (\theta
(x_{1},x_{2},x_{3},x_{4},x_{5}),
C_{x_{6}+1})\) , where \(2 \leq
x_{1}\leq x_{2} \leq x_{3} \leq x_{4}
\leq x_{5}\) and \(2 \leq
x_{6}\).
\(\underline{\mathbf{ Case\
A}}\) \(B \in g_{e} (\theta
(x_{1},x_{2},x_{3}), C_{x_{4}+1},
C_{x_{5}+1},C_{x_{6}+1})\) , where \(2
\leq x_{1}\leq x_{2} \leq
x_{3}\) and \(2 \leq
x_{4},x_{5},x_{6}\).
As \(A \cong \theta
(r,r,r,s,t,u)\) and \(B \in g_{e}
(\theta (x_{1},x_{2},x_{3}),
C_{x_{4}+1}, C_{x_{5}+1},C_{x_{6}+1})\), then by Theorem 7, we
have
\(Q(A)= z(z^{r}-1)^{2}(z^{s}-1)^{2}(z^{t}-1)(z^{u}-1)-
(z^{r}-z)^{2}(z^{s}-z)^{2}(z^{t}-z)(z^{u}-z)\).
\(Q(B)=z(z^{x_{1}}-1)(z^{x_{2}}-1)(z^{x_{3}}-1)(z^{x_{4}}-1)(z^{x_{5}}-1)(z^{x_{6}}-1)-
(z^{x_{1}}-z)(z^{x_{2}}-z)(z^{x_{3}}-z)(z^{x_{4}}-1)(z^{x_{5}}-1)(z^{x_{6}}-1)\).
Let \(Q_1(A)\) is a new polynomial
obtained by comparing \(Q(A)=
Q(B)\).
\(Q_{1}(A)=
z^{2r+2s+1}+z^{2r+t+u+1}+2z^{2r+s+u+1}+2z^{2r+s+t+1}+2z^{2r+u+t+1}+z^{2r+1}+z^{2s+1}+z^{t+u+1}+2z^{s+t+1}+2z^{s+u+1}
+2z^{r+s+1}+2z^{r+u+1}+2z^{r+2s+t+1}+2z^{r+2s+u+1}+4z^{r+s+t+u+1}+2z^{s+t+u+3}+2z^{r+t+u+3}+2z^{r+2s+t+u+1}+z^{2r+t+3}+z^{2s+t+3}
z^{t+5}+4z^{r+s+t+3}+z^{2r+u+3}+z^{2s+u+3}z^{u+5}+4z^{r+s+u+3}+z^{r+s+3}+z^{r+2s+3}+2z^{s+5}+2z^{r+5}+4z^{r+s+1}-(z^{2r+t+1}+
z^{2r+s+1}+z^{2r+u+1}+z^{2s+t+1}+z^{2s+u+1}+z^{t+1}+2z^{s+1}+2z^{s+t+u+1}+2z^{r+1}+z^{u+1}+2z^{r+2s+t+u+1}+2z^{r+t+u+1}+4z^{r+s+t+1}
+4z^{r+s+u+1}+2z^{r+2s+1}+z^{2r+2s+2}+z^{2r+t+u+2}+z^{2r+4}+2z^{2r+s+t+2}+2z^{2r+s+u+2}+2z^{2r+t+u+2}+z^{2s+4}+z^{t+u+4}+2z^{s+t+4}
+2z^{s+u+4}+2z^{r+t+4}+2z^{r+u+4}+2z^{r+2s+t+2}+2z^{r+2s+u+2}+4z^{r+s+t+u+2}+4z^{r+s+4}+z^{6})\).
\(Q_{1}(B)= z^{x_{1}+ x_{2}+ x_{3}+ x_{4}+
x_{5}} + z^{x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{6}}+ z^{x_{1}+ x_{2}+ x_{3}+
x_{4}+1} + z^{x_{1}+ x_{2}+ x_{3}+ x_{5}+ x_{6}}
+ z^{x_{1}+ x_{2}+ x_{3}+ x_{5}+1} + z^{x_{1}+ x_{2}+ x_{3}+
x_{6}+1}+ z^{x_{1}+ x_{2}+ x_{3}}+ z^{x_{1}+ x_{4}+ x_{5}+ x_{6}+1}+
z^{x_{1}+ x_{4}+ x_{5}+ 2}+ z^{x_{1}+ x_{4}+ x_{6}+ 2}+ z^{x_{1}+
x_{5}+ x_{6}+ 2}+ z^{x_{1}+ x_{4}+1}+ z^{x_{1}+ x_{5}+1}+
z^{x_{1}+ x_{6}+1}+ z^{x_{1}+ 2}+ z^{x_{2}+ x_{4}+ x_{5}+ x_{6}+1}+
z^{x_{2}+x_{4}+ x_{5}+ 2}+ z^{x_{2}+x_{4}+ x_{6}+ 2}+
z^{x_{2}+x_{4}+1}+ z^{x_{2}+ x_{5}+ x_{6}+ 2}+ z^{x_{2}+ x_{5}+1}+
z^{x_{2}+ x_{6}+1}+ z^{x_{2}+ 2}+ z^{x_{3}+ x_{4}+ x_{5}+ x_{6}+1}+
z^{x_{3}+x_{4}+ x_{5}+ 2}+ z^{x_{3}+x_{4}+ x_{6}+ 2}+
z^{x_{3}+x_{4}+1}+ z^{x_{3}+ x_{5}+ x_{6}+ 2}+
z^{x_{3}+ x_{5}+1}+z^{x_{3}+ x_{6}+1}+ z^{x_{3}+ 2}+ z^{x_{4}+x_{5}+
x_{6}+
3}+ z^{x_{4}+ x_{5}+1}+ z^{x_{4}+ x_{6}+1}+ z^{x_{4}+3}+ z^{x_{5}+
x_{6}+1}+ z^{x_{5}+3}+ z^{x_{6}+3}- ( z^{x_{1}+ x_{2}+ x_{3}+ x_{4}+
x_{5}+1} + z^{x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{6}+1}+ z^{x_{1}+ x_{2}+
x_{3}+ x_{4}} + z^{x_{1}+ x_{2}+ x_{3}+ x_{5}+ x_{6}+1}
+ z^{x_{1}+ x_{2}+ x_{3}+ x_{5}} + z^{x_{1}+ x_{2}+ x_{3}+ x_{6}}+
z^{x_{1}+ x_{2}+ x_{3}+1}+ z^{x_{1}+ x_{4}+ x_{5}+ x_{6}+2}+
z^{x_{1}+ x_{4}+ x_{5}+ 1}+ z^{x_{1}+ x_{4}+ x_{6}+ 1}+ z^{x_{1}+
x_{5}+ x_{6}+ 1}+ z^{x_{1}+ x_{4}+2}+ z^{x_{1}+ x_{5}+2}+
z^{x_{1}+ x_{6}+2}+ z^{x_{1}+1}+ z^{x_{2}+ x_{4}+ x_{5}+ x_{6}+2}+
z^{x_{2}+x_{4}+ x_{5}+ 1}+ z^{x_{2}+x_{4}+ x_{6}+ 1}+
z^{x_{2}+x_{4}+2}+ z^{x_{2}+ x_{5}+ x_{6}+ 2}+ z^{x_{2}+ x_{5}+2}+
z^{x_{2}+ x_{6}+2}+ z^{x_{2}+ 1}+ z^{x_{3}+ x_{4}+ x_{5}+ x_{6}+2}+
z^{x_{3}+x_{4}+ x_{5}+ 1}+ z^{x_{3}+x_{4}+ x_{6}+ 1}+
z^{x_{3}+x_{4}+2}+ z^{x_{3}+ x_{5}+ x_{6}+ 1}+
z^{x_{3}+ x_{5}+2}+z^{x_{3}+ x_{6}+2}+ z^{x_{3}+ 1}+ z^{x_{4}+x_{5}+
x_{6}+
1}+ z^{x_{4}+ x_{5}+3}+ z^{x_{4}+ x_{6}+3}+ z^{x_{4}+1}+ z^{x_{5}+
x_{6}+3}+ z^{x_{5}+1}+ z^{x_{6}+1} + z^{3} )\).
Compare the l.r.p in \(Q_{1}(A)\) and
the l.r.p in \(Q_{1}(B)\). Thus, \(r = 2\). Therefore, \(g(A)= g(B)= 2r\) = \(4\). Since \(A\) has one cycle of length four, therefore
\(B\) has one cycle of length \(4\). Without loss of generality, we have
four cases to consider.
1. \(x_{4}= x_{5}= x_{6}= 3\) or
2. \(x_{4}= x_{5}= 3 , x_{6} \neq 3\)
or
3. \(x_{4}= 3 , x_{5} \neq 3 ,x_{6} \neq
3\) or
4. \(x_{4}\neq 3 , x_{5} \neq 3 ,x_{6} \neq
3\).
\(\underline{\mathbf{Case\ 1}:}\)
\(x_{4}= x_{5}= x_{6}= 3\).
Therefore, \(B\) has at least three
cycles of length \(4\). While \(A\) has one cycle of the same length, by
Theorem 8 a contradiction.
\(\underline{\mathbf{Case\ 2}:}\)
\(x_{4}= x_{5}=3\), \(x_{6}\neq 3\)
Therefore, \(B\) has at least two
cycles of length \(4\). While \(A\) has one cycle of the same length, by
Theorem 8 this is a contradiction.
\(\underline{\mathbf{Case\ 3}:}\)
\(x_{4}=3\), \(x_{5}\neq3\), \(x_{6}\neq 3\) \[2s + t + u + 1 = x_{1} + x_{2} + x_{3} + x_{5} +
x_{6}.\tag{4}\]
\(Q_{2}(A)=3z^{2s+5}+3z^{t+u+5}+6z^{s+u+5}+6z^{s+t+5}+z^{t+u+1}+z^{2s+1}
+2z^{s+t+1}+2z^{s+u+1}+3z^{2s+u+3}+3z^{2s+t+3}+2z^{t+3}+2z^{u+3}+4z^{s+3}+6z^{s+t+u+3}+z^{t+7}+z^{u+7}+2z^{7}+z^{5}
-(z^{2s+t+1}+z^{2s+u+1}+z^{t+1}+2z^{s+1}+2z^{s+t+u+1}+2z^{t+u+3}+4z^{s+t+3}+4z^{s+u+3}+z^{u+1}+2z^{2s+3}+z^{2s+6}+z^{t+u+6}+2z^{s+t+6}
+2z^{s+u+6}+z^{2s+4}+z^{t+u+4}+2z^{s+t+4}+2z^{s+u+4}+2z^{2s+t+4}+2z^{2s+u+4}+2z^{t+6}+2z^{u+6}+4z^{s+6}+4z^{s+t+u+3}+z^{8}+z^{6})\).
\(Q_{2}(B)= z^{x_{1}+ x_{2}+ x_{3}+ x_{5}+3} +
z^{x_{1}+ x_{2}+ x_{3}+ x_{6}+3}+ z^{x_{1}+ x_{2}+ x_{3}+4} +
z^{x_{1}+ x_{2}+ x_{3}+ x_{5}+ x_{6}}
+ z^{x_{1}+ x_{2}+ x_{3}+ x_{5}+1} + z^{x_{1}+ x_{2}+ x_{3}+
x_{6}+1}+ z^{x_{1}+ x_{2}+ x_{3}}+ z^{x_{1}+ x_{5}+ x_{6}+4}+
z^{x_{1}+ x_{5}+ 5}+ z^{x_{1}+ x_{6}+ 5}+ z^{x_{1}+ x_{5}+ x_{6}+
2}+ z^{x_{1}+4}+ z^{x_{1}+ x_{5}+1}+ z^{x_{1}+ x_{6}+1}+ z^{x_{1}+
2}+ z^{x_{2}+ x_{5}+ x_{6}+4}+ z^{x_{2}+x_{5}+5}+ z^{x_{2}+ x_{6}+
5}+ z^{x_{2}+4}+ z^{x_{2}+ x_{5}+ x_{6}+ 2}+ z^{x_{2}+ x_{5}+1}+
z^{x_{2}+ x_{6}+1}+ z^{x_{2}+ 2}+ z^{x_{3}+ x_{5}+ x_{6}+4}+
z^{x_{3}+ x_{5}+ 5}+ z^{x_{3}+ x_{6}+ 5}+ z^{x_{3}+4}+ z^{x_{3}+
x_{5}+ x_{6}+ 2}+ z^{x_{3}+ x_{5}+1}+z^{x_{3}+ x_{6}+1}+ z^{x_{3}+
2}+ z^{x_{5}+ x_{6}+ 5}+ z^{x_{5}+4}+ z^{x_{6}+4}+ z^{6}+ z^{x_{5}+
x_{6}+1}+ z^{x_{5}+3}+ z^{x_{6}+3}- ( z^{x_{1}+ x_{2}+ x_{3}+
x_{5}+4} + z^{x_{1}+ x_{2}+ x_{3}+ x_{6}+4}+ z^{x_{1}+ x_{2}+
x_{3}+3} + z^{x_{1}+ x_{2}+ x_{3}+ x_{5}+ x_{6}+1} + z^{x_{1}+
x_{2}+ x_{3}+ x_{5}} + z^{x_{1}+ x_{2}+ x_{3}+ x_{6}}+ z^{x_{1}+
x_{2}+ x_{3}+1}+ z^{x_{1}+ x_{5}+ x_{6}+5}+ z^{x_{1}+ x_{5}+ 4}+
z^{x_{1}+ x_{6}+ 4}+ z^{x_{1}+ x_{5}+ x_{6}+ 1}+ z^{x_{1}+5}+
z^{x_{1}+ x_{5}+2}+ z^{x_{1}+ x_{6}+2}+ z^{x_{1}+1}+ z^{x_{2}+
x_{5}+ x_{6}+5}+ z^{x_{2}+x_{5}+4}+ z^{x_{2}+ x_{6}+ 4}+
z^{x_{2}+5}+ z^{x_{2}+ x_{5}+ x_{6}+ 2}+ z^{x_{2}+
x_{5}+2}+ z^{x_{2}+ x_{6}+2}+ z^{x_{2}+ 1}+ z^{x_{3} x_{5}+
x_{6}+5}+ z^{x_{3}+ x_{5}+ 4}+ z^{x_{3}+ x_{6}+ 4}+ z^{x_{3}+5}+
z^{x_{3}+ x_{5}+ x_{6}+ 1}+
z^{x_{3}+ x_{5}+2}+z^{x_{3}+ x_{6}+2}+ z^{x_{3}+ 1}+ z^{x_{5}+ x_{6}+
4}+ z^{ x_{5}+6}+ z^{ x_{6}+6}+ z^{4}+ z^{x_{5}+
x_{6}+3}+ z^{x_{5}+1}+ z^{x_{6}+1} + z^{3} )\).
Compare the l.r.p in \(Q_{2}(A)\) and
the l.r.p in \(Q_{2}(B)\). We have
\(x_{1}= 2\) or \(x_{2}= 2\) or \(x_{3}= 2\)
\(\underline{\mathbf{Case\ 3.1}:}\)
\(x_{1}=2\). Then \(2\leq x_{2}\leq x_{3}\).
\(Q_{3}(B)=
z^{x_{2}+x_{3}+x_{5}+5}+z^{x_{2}+x_{3}+x_{6}+5}+z^{x_{2}+x_{3}+6}+z^{x_{2}+x_{3}+x_{5}+3}+z^{x_{2}+x_{3}+x_{6}+3}
+z^{x_{2}+x_{3}+2}+z^{x_{5}+x_{6}+6}+z^{x_{5}+7}+z^{x_{6}+7}+z^{6}
+z^{x_{5}+x_{6}+4}+z^{x_{5}+3}+z^{x_{6}+3}+z^{x_{2}+x_{5}+x_{6}+4}+z^{x_{2}+x_{5}+5}+z^{x_{2}+x_{6}+5}
+z^{x_{2}+4}+z^{x_{2}+x_{5}+x_{6}+2}+z^{x_{2}+x_{5}+1}+z^{x_{2}+x_{6}+1}+z^{x_{2}+2}+z^{x_{3}+x_{5}+x_{6}+4}+z^{x_{3}+x_{5}+5}
+z^{x_{3}+x_{6}+5}+z^{x_{3}+4}+z^{x_{3}+x_{5}+x_{6}+2}+z^{x_{3}+x_{5}+1}+z^{x_{3}+x_{6}+1}+z^{x_{3}+2}+z^{x_{5}+x_{6}+6}
+z^{x_{5}+4}+z^{x_{6}+4}+z^{6}+z^{x_{5}+x_{6}+1}+z^{x_{5}+3}+z^{x_{6}+3}-
(z^{x_{2}+x_{3}+x_{5}+6}+z^{x_{2}+x_{3}+x_{6}
+6}+z^{x_{2}+x_{3}+5}+z^{x_{2}+x_{3}+x_{5}+2}+z^{x_{2}+x_{3}+x_{6}+2}
+z^{x_{2}+x_{3}+3}+z^{x_{5}+x_{6}+7}+z^{x_{5}+6}+z^{x_{6}+6}+z^{7}
+z^{x_{5}+x_{6}+3}+z^{x_{5}+4}+z^{x_{6}+4}+z^{x_{2}+x_{5}+x_{6}+5}+z^{x_{2}+x_{5}+4}+z^{x_{2}+x_{6}+4}+z^{x_{2}+5}
+z^{x_{2}+x_{5}+x_{6}+1}+z^{x_{2}+x_{5}+2}+z^{x_{2}+x_{6}+2}+z^{x_{2}+1}+z^{x_{3}+x_{5}+x_{6}+5}+z^{x_{3}+x_{5}+4}
+z^{x_{3}+x_{6}+4}+z^{x_{3}+5}+z^{x_{3}+x_{5}+x_{6}+1}+z^{x_{3}+x_{5}+2}+z^{x_{3}+x_{6}+2}+z^{x_{3}+1}+z^{x_{5}+x_{6}+4}
+z^{x_{5}+6}+z^{x_{6}+6}+z^{x_{5}+x_{6}+3}+z^{x_{5}+1}+z^{x_{6}+1})\).
Consider the l.r.p in \(Q_{2}(A)\) and
the l.r.p in \(Q_{3}(B)\), we have
\(s = 4\) or \(t = 4\) or \(u =
4\).
\(\underline{\mathbf{Case\ 3.1.1}:}\)
\(s = 4\).
Since the coefficient of \(-z^{s+1}\)
in \(Q_{2}(A)\) is \(2\), then there shall have one \(-x^{5}\) in \(Q_{3}(B)\). Hence, we have to consider for
\(x_{2}= 4\) or \(x_{3}= 4\) or \(x_{5}= 4\) or \(x_{6}= 4\).
\(\underline{\mathbf{Case\ 3.1.1.1}:}\)
\(x_{2}=4\).
\(Q_{3}(A)=5z^{t+9}+2z^{t+5}+3z^{t+11}+2z^{t+3}+5z^{u+9}+2z^{u+5}+3z^{u+11}+2z^{u+3}+z^{t+u+5}+3z^{13}+6z^{7}+z^{9}-
(z^{t+1}+3z^{t+7}+2z^{t+10}+2z^{t+8}+2z^{t+12}+2z^{t+6}+z^{u+1}+3z^{u+7}+2z^{u+10}+2z^{u+8}+2z^{u+12}+2z^{u+6}+2z^{t+u+3}+z^{t+u+6}+z^{t+u+4}+2z^{t+u+7}
+4z^{10}+2z^{14}+2z^{11}+z^{12}+z^{8}+z^{6})\).
\(Q_{4}(B)=
z^{x_{3}+x_{5}+9}+z^{x_{3}+x_{6}+9}+z^{x_{3}+10}+z^{x_{3}+x_{5}+7}+z^{x_{3}+x_{6}+7}
+z^{x_{3}+6}+z^{x_{5}+x_{6}+6}+z^{x_{5}+7}+z^{x_{6}+7}+3z^{6}
+z^{x_{5}+x_{6}+4}+z^{x_{5}+3}+z^{x_{6}+3}+z^{x_{5}+x_{6}+8}+z^{x_{5}+9}+z^{x_{2}+x_{6}+9}
+z^{8}+z^{x_{5}+x_{6}+6}+z^{x_{5}+5}+z^{x_{6}+5}+z^{x_{3}+x_{5}+x_{6}+4}+z^{x_{3}+x_{5}+5}
+z^{x_{3}+x_{6}+5}+z^{x_{3}+4}+z^{x_{3}+x_{5}+x_{6}+2}+z^{x_{3}+x_{5}+1}+z^{x_{3}+x_{6}+1}+z^{x_{3}+2}+z^{x_{5}+x_{6}+6}
+z^{x_{5}+4}+z^{x_{6}+4}+z^{x_{5}+x_{6}+1}+z^{x_{5}+3}+z^{x_{6}+3}-
(z^{x_{3}+x_{5}+10}+z^{x_{3}+x_{6}
+10}+z^{x_{3}+9}+z^{x_{3}+x_{5}+6}+z^{x_{3}+x_{6}+6}
+z^{x_{3}+7}+z^{x_{5}+x_{6}+7}+z^{x_{5}+6}+z^{x_{6}+6}+z^{7}
+z^{x_{5}+x_{6}+3}+z^{x_{5}+4}+z^{x_{6}+4}+z^{x_{5}+x_{6}+9}+z^{x_{5}+8}+z^{x_{6}+8}+z^{9}
+z^{x_{5}+x_{6}+5}+z^{x_{5}+6}+z^{x_{6}+6}+z^{x_{3}+x_{5}+x_{6}+5}+z^{x_{3}+x_{5}+4}
+z^{x_{3}+x_{6}+4}+z^{x_{3}+5}+z^{x_{3}+x_{5}+x_{6}+1}+z^{x_{3}+x_{5}+2}+z^{x_{3}+x_{6}+2}+z^{x_{3}+1}+z^{x_{5}+x_{6}+4}
+z^{x_{5}+6}+z^{x_{6}+6}+z^{x_{5}+x_{6}+3}+z^{x_{5}+1}+z^{x_{6}+1})\).
Consider the l.r.p in \(Q_{3}(A)\) and
the l.r.p in \(Q_{4}(B)\), we have\(x_{3}=x_{5}=x_{6}=5\).
\(Q_{5}(B)=2z^{19}+2z^{17}+z^{18}+3z^{15}+z^{10}+5z^{8}+z^{6}-(3z^{20}+z^{14}+z^{12}+2z^{11}+4z^{13})\)
\(Q_{3}(A)\neq Q_{5}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.1.2}:}\)
\(x_{3}=4\).
\(Q_{6}(B)=z^{x_{2}+x_{5}+9}+z^{x_{2}+x_{6}+9}+z^{x_{2}+10}+z^{x_{2}+x_{5}+7}+z^{x_{2}+x_{6}+7}
+z^{x_{2}+6}+z^{x_{5}+x_{6}+6}+z^{x_{5}+7}+z^{x_{6}+7}+3z^{6}
+z^{x_{5}+x_{6}+4}+z^{x_{5}+3}+z^{x_{6}+3}+z^{x_{2}+x_{5}+x_{6}+4}+z^{x_{2}+x_{5}+5}+z^{x_{2}+x_{6}+5}
+z^{x_{2}+4}+z^{x_{2}+x_{5}+x_{6}+2}+z^{x_{2}+x_{5}+1}+z^{x_{2}+x_{6}+1}+z^{x_{2}+2}+z^{x_{5}+x_{6}+8}+z^{x_{5}+9}
+z^{x_{6}+9}+z^{9}+z^{x_{5}+x_{6}+6}+z^{x_{5}+5}+z^{x_{6}+5}+z^{x_{5}+x_{6}+6}
+z^{x_{5}+4}+z^{x_{6}+4}z^{x_{5}+x_{6}+1}+z^{x_{5}+3}+z^{x_{6}+3}-
(z^{x_{2}+x_{5}+10}+z^{x_{2}+x_{6}
+10}+z^{x_{2}+9}+z^{x_{2}+x_{5}+6}+z^{x_{2}+x_{6}+6}
+z^{x_{2}+7}+z^{x_{5}+x_{6}+7}+z^{x_{5}+6}+z^{x_{6}+6}+z^{7}
+z^{x_{5}+x_{6}+3}+z^{x_{5}+4}+z^{x_{6}+4}+z^{x_{2}+x_{5}+x_{6}+5}+z^{x_{2}+x_{5}+4}+z^{x_{2}+x_{6}+4}+z^{x_{2}+5}
+z^{x_{2}+x_{5}+x_{6}+1}+z^{x_{2}+x_{5}+2}+z^{x_{2}+x_{6}+2}+z^{x_{2}+1}+z^{x_{3}+x_{5}+x_{6}+5}+z^{x_{3}+x_{5}+4}
+z^{x_{6}+8}+z^{9}+z^{x_{5}+x_{6}+5}+z^{x_{5}+6}+z^{x_{6}+6}+z^{x_{5}+x_{6}+4}
+z^{x_{5}+6}+z^{x_{6}+6}+z^{x_{5}+x_{6}+3}+z^{x_{5}+1}+z^{x_{6}+1})\).
Consider the l.r.p in \(Q_{3}(A)\) is
\(-z^{6}\) and the l.r.p in \(Q_{6}(B)\) is \(3z^{6}\), since \(2 \leq x_{2}\leq 4\) and \(x_{5},x_{6}\geq 4\)
\(Q_{3}(A)\neq Q_{6}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.1.3}:}\)
\(x_{5}=4\).
\(Q_{7}(B)=
z^{x_{2}+x_{3}+9}+z^{x_{2}x_{6}+9}+z^{x_{2}+x_{3}+6}+z^{x_{2}+x_{3}+8}+z^{x_{2}+x_{3}+x_{6}+3}
+z^{x_{2}+x_{3}+2}+z^{x_{6}+10}+z^{11}+z^{x_{6}+7}+2z^{6}
+z^{x_{6}+10}+z^{7}+z^{x_{6}+3}+z^{x_{2}+x_{6}+8}+z^{x_{2}+9}+z^{x_{2}+x_{6}+5}
+z^{x_{2}+4}
+z^{x_{2}+x_{6}+6}+z^{x_{2}+5}+z^{x_{2}+x_{6}+1}+z^{x_{2}+2}+z^{x_{3}+x_{6}+8}+z^{x_{3}+9}
+z^{x_{3}+x_{6}+5}+z^{x_{3}+4}+z^{x_{3}+x_{6}+6}+z^{x_{3}+5}+z^{x_{3}+x_{6}+1}+z^{x_{3}+2}+z^{x_{6}+10}
+z^{x_{6}+4}+z^{x_{6}+5}+z^{7}+z^{x_{6}+3}-
(z^{x_{2}+x_{3}+10}+z^{x_{2}+x_{3}+x_{6}
+6}+z^{x_{2}+x_{3}+5}+z^{x_{2}+x_{3}+6}+z^{x_{2}+x_{3}+x_{6}+2}
+z^{x_{2}+x_{3}+3}+z^{x_{6}+11}+z^{10}+z^{x_{6}+6}+z^{7}
+z^{x_{6}+7}+z^{x_{6}+4}+z^{x_{2}+x_{6}+9}+z^{x_{2}+8}+z^{x_{2}+x_{6}+4}+z^{x_{2}+5}
+z^{x_{2}+x_{6}+5}+z^{x_{2}+6}+z^{x_{2}+x_{6}+2}+z^{x_{2}+1}+z^{x_{3}+x_{6}+9}+z^{x_{3}+8}
+z^{x_{3}+x_{6}+4}+z^{x_{3}+5}+z^{x_{3}+x_{6}+5}+z^{x_{3}+6}+z^{x_{3}+x_{6}+2}+z^{x_{3}+1}+z^{x_{6}+8}
+z^{10}+z^{x_{6}+6}+z^{x_{6}+7}+z^{x_{6}+1})\).
Consider the l.r.p in \(Q_{3}(A)\) and
the l.r.p in \(Q_{7}(B)\), we have
\(x_{2}=x_{3}=x_{6}=5\).
\(Q_{4}(A)=5z^{t+9}+2z^{t+5}+3z^{t+11}+2z^{t+3}+5z^{u+9}+2z^{u+5}+3z^{u+11}+2z^{u+3}+z^{t+u+5}+3z^{13}+6z^{7}+z^{9}-
(z^{t+1}+3z^{t+7}+2z^{t+10}+2z^{t+8}+2z^{t+12}+2z^{t+6}+z^{u+1}+3z^{u+7}+2z^{u+10}+2z^{u+8}+2z^{u+12}+2z^{u+6}+2z^{t+u+3}+z^{t+u+6}+z^{t+u+4}+2z^{t+u+7}
+4z^{10}+2z^{14}+2z^{11}+z^{12}+z^{8})\).
\(Q_{8}(B)=3z^{18}+z^{15}+z^{17}+z^{16}+3z^{7}+2z^{9}+2z^{8}-(z^{21}+z^{11}+4z^{12}+3z^{13}+z^{19})\).
Compare the l.r.p in \(Q_{4}(A)\) and
the l.r.p in \(Q_{8}(B)\), we
have
\(Q_{4}(A)\neq Q_{8}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.1.4}:}\)
\(x_{6}=4\).
Similar to Case 3.1.1.3, we obtain a contradiction.
\(\underline{\mathbf{Case\ 3.1.2}:}\)
\(t = 4\).
Therefore \(s = 2\) or \(s = 3\) or \(s =
4\)
If \(s = 2\), \(A \cong\theta(2,2,2,2,4,u)\) implies \(A\) is \(\chi\)-unique by Lemma 15.
If \(s = 4\), \(A\cong\theta(2,2,4,4,4,u)\) implies \(A\) is \(\chi\)-unique by Lemma 14.
If \(s = 3\).
\(Q_{5}(A)=6z^{u+9}+3z^{u+8}+z^{u+11}+z^{u+5}+2z^{u+4}+2z^{u+3}+3z^{13}+6z^{12}+4z^{11}+5z^{7}+4z^{6}-(
2z^{u+7}+4z^{u+6}+z^{u+1}+z^{u+10}+2z^{u+7}+2z^{u+6}+z^{11}+6z^{9}+7z^{10}+2z^{4}+z^{12}+2z^{14}+2z^{11}+z^{6})\).
\(Q_{9}(B)=
z^{x_{2}+x_{3}+x_{5}+5}+z^{x_{2}+x_{3}+x_{6}+5}+z^{x_{2}+x_{3}+6}+z^{x_{2}+x_{3}+x_{5}+3}+z^{x_{2}+x_{3}+x_{6}+3}
+z^{x_{2}+x_{3}+2}+z^{x_{5}+x_{6}+6}+z^{x_{5}+7}+z^{x_{6}+7}+z^{6}
+z^{x_{5}+x_{6}+4}+z^{x_{5}+3}+z^{x_{6}+3}+z^{x_{2}+x_{5}+x_{6}+4}+z^{x_{2}+x_{5}+5}+z^{x_{2}+x_{6}+5}
+z^{x_{2}+4}+z^{x_{2}+x_{5}+x_{6}+2}+z^{x_{2}+x_{5}+1}+z^{x_{2}+x_{6}+1}+z^{x_{2}+2}+z^{x_{3}+x_{5}+x_{6}+4}+z^{x_{3}+x_{5}+5}
+z^{x_{3}+x_{6}+5}+z^{x_{3}+4}+z^{x_{3}+x_{5}+x_{6}+2}+z^{x_{3}+x_{5}+1}+z^{x_{3}+x_{6}+1}+z^{x_{3}+2}+z^{x_{5}+x_{6}+6}
+z^{x_{5}+4}+z^{x_{6}+4}+z^{6}+z^{x_{5}+x_{6}+1}+z^{x_{5}+3}+z^{x_{6}+3}-
(z^{x_{2}+x_{3}+x_{5}+6}+z^{x_{2}+x_{3}+x_{6}
+6}+z^{x_{2}+x_{3}+5}+z^{x_{2}+x_{3}+x_{5}+2}+z^{x_{2}+x_{3}+x_{6}+2}
+z^{x_{2}+x_{3}+3}+z^{x_{5}+x_{6}+7}+z^{x_{5}+6}+z^{x_{6}+6}+z^{7}
+z^{x_{5}+x_{6}+3}+z^{x_{5}+4}+z^{x_{6}+4}+z^{x_{2}+x_{5}+x_{6}+5}+z^{x_{2}+x_{5}+4}+z^{x_{2}+x_{6}+4}+z^{x_{2}+5}
+z^{x_{2}+x_{5}+x_{6}+1}+z^{x_{2}+x_{5}+2}+z^{x_{2}+x_{6}+2}+z^{x_{2}+1}+z^{x_{3}+x_{5}+x_{6}+5}+z^{x_{3}+x_{5}+4}
+z^{x_{3}+x_{6}+4}+z^{x_{3}+5}+z^{x_{3}+x_{5}+x_{6}+1}+z^{x_{3}+x_{5}+2}+z^{x_{3}+x_{6}+2}+z^{x_{3}+1}+z^{x_{5}+x_{6}+4}
+z^{x_{5}+6}+z^{x_{6}+6}+z^{x_{5}+x_{6}+3}+z^{x_{5}+1}+z^{x_{6}+1})\).
Compare the l.r.p in \(Q_{5}(A)\) and
the l.r.p in \(Q_{9}(B)\), we have
\(x_{2}= x_{3}=3\).
\(Q_{6}(A)=6z^{u+9}+3z^{u+8}+z^{u+11}+z^{u+5}+2z^{u+4}+2z^{u+3}+3z^{13}+6z^{12}+4z^{11}+5z^{7}+z^{6}-(
2z^{u+7}+4z^{u+6}+z^{u+1}+z^{u+10}+2z^{u+7}+2z^{u+6}+z^{11}+6z^{9}+7z^{10}+z^{12}+2z^{14}+2z^{11})\).
\(Q_{10}(B)=z^{x_{5}+x_{6}+6}+z^{x_{5}+x_{6}+4}+2z^{x_{5}+x_{6}+7}+2z^{x_{5}+x_{6}+5}+z^{x_{5}+x_{6}+6}+z^{x_{5}+x_{6}+1}
+z^{x_{5}+11} +z^{x_{5}+9}+2z^{x_{5}+3}+2z^{x_{5}+8}
+2z^{x_{5}+4}+z^{x_{6}+11} +z^{x_{6}+9}+2z^{x_{6}+3}+2z^{x_{6}+8}
+2z^{x_{6}+4}+z^{12}+2z^{7}+2z^{5}-(2z^{x_{5}+x_{6}+8}+2z^{x_{5}+x_{6}+3}+2z^{x_{5}+x_{6}+7}+3z^{x_{5}+x_{6}+4}+
+z^{x_{5}+12} +2z^{x_{5}+6}+2z^{x_{5}+5}+z^{x_{5}+8}
+z^{x_{5}+7}+z^{x_{5}+1}+z^{x_{6}+12} +z^{x_{6}+8}+2z^{x_{6}+5}
+2z^{x_{6}+7}+z^{x_{6}+6}+z^{x_{6}+1}+z^{11}+z^{9}+z^{7}+z^{8})\).
Compare the l.r.p in \(Q_{6}(A)\) and
the l.r.p in \(Q_{10}(B)\), we have u =
5, \(x_{5}= x_{6}=4\).
\(Q_{7}(A)=z^{16}+4z^{14}+4z^{13}+z^{12}+2z^{9}+3z^{8}-(5z^{11}+z^{15}+6z^{10}+6z^{9}+2z^{14})\)
\(Q_{11}(B)=4z^{15}+2z^{14}+4z^{13}+6z^{12}-(4z^{16}+2z^{15}+6z^{11}+5z^{12}+3z^{10}+z^{8})\)
Compare the l.r.p in \(Q_{7}(A)\) and
the l.r.p in \(Q_{11}(B)\), we
have
\(Q_{7}(A)\neq Q_{11}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.3}:}\)
\(u=4\).
Therefore, we have \(2\leq s\leq t\leq
4\).
If \(s = 2\), then \(t = 2\) or \(t =
3\) or \(t = 4\)
\(s = 2\), \(t = 2\), \(A\cong\theta(2,2,2,2,2,4)\), \(A\) is \(\chi\)-unique, by Lemma 9.
\(s = 2\), \(t = 3\), \(A\cong\theta(2,2,2,2,3,4)\), \(A\) is \(\chi\)-unique by Lemma 15.
\(s = 2\), \(t = 4\), \(A\cong\theta(2,2,2,2,4,4)\), \(A\) is \(\chi\)-unique, by Lemma 9.
If \(s = 3\),
\(s = 3\), \(t = 3\), \(A\cong\theta(2,2,3,3,3,4)\), \(A\) is \(\chi\)-unique, by Lemma 14.
\(s = 3\), \(t = 4\), \(A\cong\theta(2,2,3,3,4,4)\), \(A\) is \(\chi\)-unique, by Lemma 13.
If \(s = 4\)
\(s = 4\), \(t = 4\), \(A\cong\theta(2,2,4,4,4,4)\), \(A\) is \(\chi\)-unique by Lemma 9.
\(\underline{\mathbf{Case\ 3.2}:}\)
\(x_{2}=2\).
Then \(x_{1}=2\). Hence \(B\) has at least two cycles of length \(4\), a contradiction.
\(\underline{\mathbf{Case\ 3.3}:}\)
\(x_{3}=2\). Then \(x_{1}=x_{2}=2\). Hence \(B\) has at least four cycles of length
\(4\), a contradiction.
\(\underline{\mathbf{Case\ 4}:}\)
\(x_{4}\neq 3, x_{5}\neq 3, x_{6}\neq
3\)
We know that \(x_{4}, x_{5}, x_{6}
>3\). Given that \(B\) shall
has one cycle of length \(4\), then
\(x_{1}+x_{2}= 4\), \(x_{1}=x_{2}= 2\).
Then \(s = 2\) or \(t = 2\) or \(u =
2\) .
If \(s = 2\), \(A\cong\theta(2,2,2,2,t,u)\), \(A\) is \(\chi\)-unique by Lemma 15.
If \(t = 2\), \(A\cong\theta(2,2,2,2,2,u)\), \(A\) is \(\chi\)-unique, by Lemma 9.
If \(u = 2\), \(A\cong\theta(2,2,2,2,2,2)\), \(A\) is \(\chi\)-unique, by Theorem 1.
\(\underline{\mathbf{Case\ B}:}\)
\(B \in g_{e} (\theta
(x_{1},x_{2},x_{3},x_{4}),
C_{x_{5}+1},C_{x_{6}+1})\), where \(2
\leq x_{1}\leq x_{2} \leq
x_{3} \leq x_{4 }\) and \(2 \leq
x_{5},x_{6}\).
As \(A\) \(\cong \theta(r,r,s,s,t,u)\) and \(B\) \(\in g_{e}
(\theta (x_{1},x_{2},x_{3},x_{4}),
C_{x_{5}+1},C_{x_{6}+1})\), then by Theorem 1, we have
\(Q_{8}(A)\)= \(z(z^{r}-1)^{2}(z^{s}-1)^{2}(z^{t}-1)(z^{u}-1)-
(z^{r}-z)^{2}(z^{s}-z)^{2}(z^{t}-z)(z^{u}-z)\).
\(Q_{12}(B)\)= \(z(z^{x_{1}}-1)(z^{x_{2}}-1)(z^{x_{3}}-1)(z^{x_{4}}-1)(z^{x_{5}}-1)(z^{x_{6}}-1)-
(z^{x_{1}}-z)(z^{x_{2}}-z)(z^{x_{3}}-z)(z^{x_{4}}-z)(z^{x_{5}}-1)(z^{x_{6}}-1)\).
\(Q_{8}(A)= Q_{12}(B)\), yields
\(Q_{9}(A)=
z^{2r+2s+1}+z^{2r+t+u+1}+2z^{2r+s+u+1}+2z^{2r+s+t+1}+2z^{2r+u+t+1}+z^{2r+1}+z^{2s+1}+z^{t+u+1}+2z^{s+t+1}+2z^{s+u+1}
+2z^{r+s+1}+2z^{r+u+1}+2z^{r+2s+t+1}+2z^{r+2s+u+1}+4z^{r+s+t+u+1}+2z^{s+t+u+3}+2z^{r+t+u+3}+2z^{r+2s+t+u+1}+z^{2r+t+3}+z^{2s+t+3}
z^{t+5}+4z^{r+s+t+3}+z^{2r+u+3}+z^{2s+u+3}z^{u+5}+4z^{r+s+u+3}+z^{r+s+3}+z^{r+2s+3}+2z^{s+5}+2z^{r+5}+4z^{r+s+1}-(z^{2r+t+1}+
z^{2r+s+1}+z^{2r+u+1}+z^{2s+t+1}+z^{2s+u+1}+z^{t+1}+2z^{s+1}+2z^{s+t+u+1}+2z^{r+1}+z^{u+1}+2z^{r+2s+t+u+1}+2z^{r+t+u+1}+4z^{r+s+t+1}
+4z^{r+s+u+1}+2z^{r+2s+1}+z^{2r+2s+2}+z^{2r+t+u+2}+z^{2r+4}+2z^{2r+s+t+2}+2z^{2r+s+u+2}+2z^{2r+t+u+2}+z^{2s+4}+z^{t+u+4}+2z^{s+t+4}
+2z^{s+u+4}+2z^{r+t+4}+2z^{r+u+4}+2z^{r+2s+t+2}+2z^{r+2s+u+2}+4z^{r+s+t+u+2}+4z^{r+s+4}+z^{6})\).
\(Q_{13}(B)= z^{x_{1}+ x_{2}+ x_{3}+ x_{4}+
x_{5}} + z^{x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{6}}+ z^{x_{1}+ x_{2}+ x_{3}+
x_{4}+1} + z^{x_{1}+ x_{2}+ x_{5}+ x_{6}+1}
+ z^{x_{1}+ x_{2}+ x_{5}+2} + z^{x_{1}+ x_{2}+ x_{6}+2}+ z^{x_{1}+
x_{2}+1}+ z^{x_{1}+ x_{3}+ x_{5}+ x_{6}+1}+ z^{x_{1}+ x_{3}+ x_{5}+
2}+ z^{x_{1}+ x_{3}+ x_{6}+ 2}+ z^{x_{1}+ x_{3}+ 1}+ z^{x_{1}+
x_{4}+ x_{5}+ x_{6}+1}+ z^{x_{1}+ x_{4}+ x_{5}+ 2}+ z^{x_{1}+ x_{4}+
x_{6}+ 2}+ z^{x_{1}+ x_{4}+ 1}+ z^{x_{1}+ x_{5}+ x_{6}+ 3}+
z^{x_{1}+ x_{5}+1}+ z^{x_{1}+ x_{6}+1}+ z^{x_{1}+ 3}+ z^{x_{2}+
x_{3}+ x_{5}+ x_{6}+1}+ z^{x_{2}+x_{3}+ x_{5}+ 2}+
z^{x_{2}+x_{3}+x_{6}+2}+z^{x_{2}+ x_{3}+ 1}+ z^{x_{2}+ x_{4}+ x_{5}+
x_{6}+1}+ z^{x_{2}+x_{4}+ x_{5}+ 2}+ z^{x_{2}+x_{4}+ x_{6}+ 2}+
z^{x_{2}+x_{4}+1}+ z^{x_{2}+ x_{5}+ x_{6}+ 3}+ z^{x_{2}+ x_{5}+1}+
z^{x_{2}+ x_{6}+1}+ z^{x_{2}+ 3}+ z^{x_{3}+ x_{4}+ x_{5}+ x_{6}+1}+
z^{x_{3}+x_{4}+ x_{5}+ 2}+ z^{x_{3}+x_{4}+ x_{6}+ 2}+
z^{x_{3}+x_{4}+1}+ z^{x_{3}+ x_{5}+ x_{6}+ 2}+
z^{x_{3}+ x_{5}+1}+z^{x_{3}+ x_{6}+1}+ z^{x_{3}+ 3}+ z^{x_{4}+x_{5}+
x_{6}+
3}+ z^{x_{4}+ x_{5}+1}+ z^{x_{4}+ x_{6}+1}+ z^{x_{4}+3}+ z^{x_{5}+
x_{6}+1}+ z^{x_{5}+4}+ z^{x_{6}+4}- ( z^{x_{1}+ x_{2}+ x_{3}+ x_{4}+
x_{5}+1} + z^{x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{6}+1}+ z^{x_{1}+ x_{2}+
x_{3}+ x_{4}} + z^{x_{1}+ x_{2}+ x_{3}+ x_{5}+ x_{6}+2}
+ z^{x_{1}+ x_{2}+ x_{5}+1} + z^{x_{1}+ x_{2}+ x_{6}+1}+ z^{x_{1}+
x_{2}+2}+ z^{x_{1}+ x_{3}+ x_{5}+ x_{6}+2}+ z^{x_{1}+ x_{3}+ x_{5}+
1}+ z^{x_{1}+ x_{3}+ x_{6}+ 1}++ z^{x_{1}+ x_{3}+2} +z^{x_{1}+
x_{4}+ x_{5}+ x_{6}+2}+ z^{x_{1}+ x_{4}+ x_{5}+ 1}+ z^{x_{1}+
x_{4}+ x_{6}+ 1}+ z^{x_{1}+ x_{5}+ x_{6}+ 1}+ z^{x_{1}+ x_{4}+2}+
z^{x_{1}+ x_{5}+3}+ z^{x_{1}+ x_{6}+3}+ z^{x_{1}+1}+z^{x_{2}+ x_{3}+
x_{5}+ x_{6}+2}+ z^{x_{2}+x_{3}+ x_{5}+ 1}+ z^{x_{2}+x_{3}+ x_{6}+
1}+ z^{x_{2}+x_{3}+2}+ z^{x_{2}+x_{4}+ x_{5}+ x_{6}+2}+
z^{x_{2}+x_{4}+ x_{5}+ 1}+ z^{x_{2}+x_{4}+ x_{6}+ 1}+
z^{x_{2}+x_{4}+2}+ z^{x_{2}+ x_{5}+ x_{6}+ 1}+ z^{x_{2}+ x_{5}+3}+
z^{x_{2}+ x_{6}+3}+ z^{x_{2}+ 1}+ z^{x_{3}+ x_{4}+ x_{5}+ x_{6}+2}+
z^{x_{3}+x_{4}+ x_{5}+ 1}+ z^{x_{3}+x_{4}+ x_{6}+ 1}+
z^{x_{3}+x_{4}+2}+ z^{x_{3}+ x_{5}+ x_{6}+ 1}+ z^{x_{3}+
x_{5}+3}+z^{x_{3}+ x_{6}+3}+ z^{x_{3}+ 1}+ z^{x_{4}+x_{5}+ x_{6}+
1}+ z^{x_{4}+ x_{5}+3}+ z^{x_{4}+ x_{6}+3}+ z^{x_{4}+1}+ z^{x_{5}+
x_{6}+4}+ z^{x_{5}+1}+ z^{x_{6}+1} + z^{4} )\).
Since \(2\leq r\leq s\leq t\leq u\).
Therefore, by comparing the l.r.p in \(Q_{9}(A)\) and the l.r.p in \(Q_{13}(B)\), we have \(r =2\) or \(r =
3\).
\(\underline{\mathbf{Case\ 1}:}\)
\(r = 2\)
Then \(g(A)= g(B)= 2r =4\). Since \(A\) has one cycle of length four, therefore
\(B\) has one cycle of length \(4\). Without loss of generality, we have
three cases to consider ,
1. \(x_{5}= x_{6}= 3\) or
2. \(x_{5}= 3 , x_{6} \neq 3\) or
3. \(x_{5} \neq 3 ,x_{6} \neq 3\)
\(\underline{\mathbf{Case\ 1.1}:}\)
\(x_{5}= x_{6}= 3\).
Since \(B\) has at least two cycles of
length \(4\), a contradiction.
\(\underline{\mathbf{Case\ 1.2}:}\)
\(x_{5}= 3\), \(x_{6}\neq 3\)
We know that \(x_{6} > 3\).
Substituting into \(Q_{9}(A)\) and
\(Q_{13}(B)\). We obtain that there is
\(-2z^{3}\) in \(Q_{9}(A)\). Hence there are six cases to be
considered, that are \(x_{1}=x_{2}=2\)
or \(x_{1}=x_{3}=2\) or \(x_{1}=x_{4}=2\) or \(x_{2}=x_{3}=2\) or \(x_{3}=x_{4}=2\) or \(x_{2}=x_{4}=2\).
For \(x_{1}=x_{2}=2\), \(B\) has at least two cycles of length \(4\), a contradiction. \(B\) has at least three cycles of length
\(4\) for all other cases. Thus, a
contradiction.
\(\underline{\mathbf{Case\ 1.3}:}\)
\(2\leq x_{5}\leq x_{6}\)
We know that \(x_{5}, x_{6} > 3\).
Hence \(x_{1}+x_{2}=4\), implying \(x_{1}=x_{2}=2\).
Considering the l.r.p in \(Q_{9}(A)\)
and the l.r.p in \(Q_{13}(B)\), we have
\(s = 3\) or \(t = 3\) or \(u =
3\).
\(\underline{\mathbf{Case\ 1.3.1}:}\)
\(s = 3\).
Note that the term \(-z^{s+1}\) in
\(Q_{9}(A)\) has coefficient \(2\), then \(x_{3}=3\) or \(x_{4}=3\).
\(\underline{\mathbf{Case\ 1.3.1.1}:}\)
\(x_{3}=3\)
\(Q_{14}(B)=z^{x_{4}+x_{5}+7}+z^{x_{4}+x_{6}+7}+z^{x_{4}+8}+z^{x_{5}+x_{6}+5}+z^{x_{5}+6}+z^{x_{6}+6}+2z^{5}+3z^{x_{5}+x_{6}+6}+2z^{x_{5}+7}
+2z^{x_{6}+7}+3z^{x_{4}+x_{5}+x_{6}+3}+2z^{x_{4}+x_{5}+4}+2z^{x_{4}+x_{6}+4}+3z^{x_{4}+3}+2z^{x_{5}+x_{6}+5}
+2z^{x_{5}+3}+2z^{x_{6}+3}+z^{x_{4}+x_{5}+x_{6}+4}+z^{x_{4}+x_{5}+5}+z^{x_{4}+x_{6}+5}+z^{x_{4}+4}+2z^{x_{5}+4}+2z^{x_{6}+4}+z^{x_{4}+x_{5}+1}+z^{x_{4}+x_{6}+1}+z^{x_{5}+x_{6}+1}-
(z^{x_{4}+x_{5}+8}+z^{x_{4}+x_{6}+8}+z^{x_{4}+7}+z^{x_{5}+x_{6}+6}+3z^{x_{5}+5}+3z^{x_{6}+5}+z^{6}+z^{x_{5}+x_{6}+7}+3z^{x_{5}+6}
+3z^{x_{6}+6}+2z^{7}+2z^{x_{4}+x_{5}+x_{6}+4}
+3z^{x_{4}+x_{5}+3}+3z^{x_{4}+x_{6}+3}+2z^{x_{4}+4}+2z^{x_{5}+x_{6}+3}+z^{x_{5}+x_{6}+7}+z^{x_{4}+x_{5}+x_{6}+5}+z^{x_{4}+x_{5}+4}
+z^{x_{4}+x_{6}+4}+z^{x_{4}+5}+2z^{x_{5}+x_{6}+4}+z^{x_{4}+x_{5}+x_{6}+1}+z^{x_{4}+1}+z^{x_{5}+1}+z^{x_{6}+1}).\)
Considering the l.r.p in \(Q_{9}(A)\)
and the l.r.p in \(Q_{14}(B)\), we have
\(x_{4}= x_{5}=4\), or \(x_{4}= x_{6}=4\), or \(x_{5}= x_{6}=4\). .
\(\underline{\mathbf{Case\ 1.3.1.1.1}:}\)
\(x_{4}= x_{5}=4\).
\(Q_{15}(B)=z^{15}+3z^{x_{6}+11}+2z^{12}+3z^{x_{6}+9}+2z^{x_{6}+10}+2z^{x_{6}+3}
+z^{x_{6}+12}+z^{13}+2z^{8}+2z^{x_{6}+4}-(z^{16}+3z^{x_{6}+6}+3z^{11}+3z^{9}+z^{x_{6}+5}+z^{6}+2z^{10}+2z^{x_{6}+6}+3z^{x_{6}+7}+z^{+13}+z^{x_{6}+8}+z^{x_{6}+1}).\)
Compare the l.r.p in \(Q_{9}(A)\) and
the l.r.p in \(Q_{15}(B)\), we have
\(t = 5\) or \(u = 5\).
If \(u = 5\) \(3\leq t \leq 5\)
If \(t = 3\), \(A\cong\theta(2,2,3,3,3,5)\), \(A\) is \(\chi\)-unique by Lemma 14.
If \(t = 4\). \(Q_{10}(A)=4z^{14}+6z^{12}+7z^{13}+z^{10}+z^{15}+3z^{11}-(3z^{13}+5z^{12}+7z^{10}+8z^{11}+2z^{14}+2z^{15}).\)
\(Q_{16}(B)=z^{15}+3z^{x_{6}+11}+2z^{12}+3z^{x_{6}+9}+2z^{x_{6}+10}+2z^{x_{6}+3}
+z^{x_{6}+12}+z^{13}+2z^{8}+2z^{x_{6}+4}-(z^{16}+3z^{x_{6}+6}+3z^{11}+3z^{9}+z^{x_{6}+5}+z^{6}+2z^{10}+2z^{x_{6}+6}+3z^{x_{6}+7}+z^{+13}+z^{x_{6}+8}+z^{x_{6}+1}).\)
Compare the l.r.p in \(Q_{10}(A)\) and
the l.r.p in \(Q_{16}(B)\), we have
\(x_{6} = 4\).
\(Q_{11}(A)=4z^{14}+6z^{12}+7z^{13}+z^{10}+z^{15}+3z^{11}-(3z^{13}+5z^{12}+7z^{10}+8z^{11}+2z^{14}+2z^{15}).\)
\(Q_{17}(B)=5z^{15}+5z^{13}+2z^{14}-(3z^{16}+6z^{11}+z^{15}+4z^{10}+z^{13}).\)
\(Q_{11}(A)\neq Q_{17}(B)\), a
contradiction.
If \(t = 5\), \(A\cong\theta(2,2,3,3,5,5)\), \(A\) is \(\chi\)-unique by Lemma 13.
\(\underline{\mathbf{Case\ 1.3.1.1.2}:}\)
\(x_{4}= x_{6}=4\).
Similarly to Case 1.3.1.1.1, we obtain a contradiction.
\(\underline{\mathbf{Case\ 1.3.1.1.3}:}\)
\(x_{5}= x_{6}=4\).
Similarly to Case 1.3.1.1.1, we obtain a contradiction.
\(\underline{\mathbf{Case\ 1.3.1.2}:}\)
\(x_{4}= 3\).
Similarly to Case 1.3.1.1.1, we obtain a contradiction.
\(\underline{\mathbf{Case\ 1.3.2}:}\)
\(t = 3\).
Since \(2\leq s\leq 3\). We know that
\(s = 2\) or \(s = 3\).
If \(s = 2\), \(A\cong\theta(2,2,2,2,3,u)\), \(A\) is \(\chi\)-unique by Lemma 15.
If \(s = 3\), \(A\cong\theta(2,2,3,3,3,u)\), \(A\) is \(\chi\)-unique by Lemma 14.
\(\underline{\mathbf{Case\ 1.3.3}:}\)
\(u = 3\). \(2\leq s\leq t\leq 3\). We know that \(s = t = 2\) or \(s = 2\), \(t =
3\) or \(s = t = 3\).
If \(s = t = 2\), \(A\cong\theta(2,2,2,2,2,u)\), \(A\) is \(\chi\)-unique by Lemma 8.
If \(s = 2\), \(t = 3\), \(A\cong\theta(2,2,2,2,3,u)\), \(A\) is \(\chi\)-unique by Lemma 15.
If \(s = t = 3\), \(A\cong\theta(2,2,3,3,3,u)\), \(A\) is \(\chi\)-unique by Lemma 14.
\(\underline{\mathbf{Case\ 2}:}\)
\(r = 3\).
Then \(g(A)= g(B)= 2r = 6\). Since
\(A\) has one cycle of length \(6\), therefore \(B\) has one cycle of length \(6\). Without loss of generality, we have
three cases to consider,
1. \(x_{5}= x_{6}= 5\) or
2. \(x_{5}= 5 , x_{6} \neq 5\) or
3. \(x_{5} \neq 5 ,x_{6} \neq 5\)
\(\underline{\mathbf{Case\ 2.1}:}\)
\(x_{5}= x_{6}= 5\).
\(B\) has at least two cycles of length
\(6\), a contradiction.
\(\underline{\mathbf{Case\ 2.2}:}\)
\(x_{5}= 5\) , \(x_{6} \neq 5\).
compare the l.r.p in \(Q_{10}(A)\) and
\(Q_{21}(B)\), we have a
contradiction.
\(\underline{\mathbf{Case\ 2.3}:}\)
\(x_{5} \neq 5\), \(x_{6} \neq 5\).
We know that \(x_{5}, x_{6}\geq 5\).
Hence \(x_{1}+ x_{2}= 6\).
\(x_{1}= x_{2}= 3\).
\(Q_{11}(A)=z^{2s+7}+z^{t+u+7}+z^{7}+2z^{s+u+7}+2z^{s+t+7}+z^{t+u+1}+z^{2s+1}+2z^{s+t+1}+2z^{s+u+1}+2z^{2s+t+4}+2z^{2s+u+4}
+2z^{t+4}+2z^{u+4}+4z^{s+t+u+4}+2z^{s+t+u+3}+2z^{s+u+6}+z^{t+9}+z^{2s+t+3}+z^{t+5}+4z^{s+t+6}+z^{u+9}+z^{2s+u+3}+z^{u+5}+4z^{s+u+6}+2z^{s+6}+2z^{s+5}+2z^{8}+2z^{2s+6}+4z^{s+4}
-(3z^{t+7}+3z^{u+7}+5z^{s+7}+z^{2s+t+1}+z^{2s+u+1}+z^{t+1}+2z^{s+t+u+1}+4z^{s+t+4}+4z^{s+u+4}+z^{u+1}
+2z^{2s+4}+2z^{s+1}+z^{t+u+4}+2z^{2s+8}+z^{t+u+8}
+z^{10}+2z^{2s+t+u+2}+2z^{s+t+8}+2z^{s+u+8}+z^{2s+4}+z^{t+u+4}+2z^{s+t+4}+2z^{s+u+4}+2z^{2s+t+5}+2z^{2s+u+5}+4z^{s+t+u+4}+z^{6})\).
\(Q_{22}(B)= z^{ x_{3}+ x_{4}+ x_{5}+6} + z^{
x_{3}+ x_{4}+ x_{6}+6}+ z^{ x_{3}+ x_{4}+7} + z^{ x_{5}+ x_{6}+7}
+ z^{ x_{5}+8} + z^{ x_{6}+8}+ z^{7}+ z^{ x_{3}+ x_{5}+ x_{6}+4}+
z^{ x_{3}+ x_{5}+ 5}+ z^{x_{3}+ x_{6}+ 5}+ z^{x_{3}+ 4}+ z^{ x_{4}+
x_{5}+ x_{6}+4}+ z^{ x_{4}+ x_{5}+ 5}+ z^{ x_{4}+ x_{6}+ 5}+ z^{
x_{4}+ 4}+ z^{ x_{5}+ x_{6}+ 6}+ z^{ x_{5}+4}+ z^{ x_{6}+4}+ z^{6}+
z^{ x_{3}+ x_{5}+ x_{6}+4}+ z^{x_{3}+ x_{5}+ 5}+
z^{x_{3}+x_{6}+5}+z^{ x_{3}+ 4}+ z^{ x_{4}+ x_{5}+ x_{6}+4}+
z^{x_{4}+ x_{5}+ 5}+ z^{x_{4}+ x_{6}+ 5}+ z^{x_{4}+4}+ z^{ x_{5}+
x_{6}+ 6}+ z^{x_{5}+4}+ z^{x_{6}+4}+ z^{6}+ z^{x_{3}+x_{4}+ x_{5}+
2}+ z^{x_{3}+x_{4}+ x_{6}+ 2}+ z^{x_{3}+x_{4}+1}+ z^{x_{3}+ x_{5}+
x_{6}+ 2}+ z^{x_{3}+ x_{5}+1}+z^{x_{3}+ x_{6}+1}+ z^{x_{3}+ 3}+
z^{x_{4}+x_{5}+ x_{6}+
3}+ z^{x_{4}+ x_{5}+1}+ z^{x_{4}+ x_{6}+1}+ z^{x_{4}+3}+ z^{x_{5}+
x_{6}+1}+ z^{x_{5}+4}+ z^{x_{6}+4}-
( z^{x_{3}+ x_{4}+ x_{5}+7} + z^{x_{3}+ x_{4}+ x_{6}+7}+ z^{ x_{3}+
x_{4}+6} + z^{x_{3}+ x_{5}+ x_{6}+8} + z^{x_{5}+7} + z^{x_{6}+7}+
z^{8}+ z^{ x_{3}+ x_{5}+ x_{6}+5}+ z^{x_{3}+ x_{5}+ 4}+ z^{ x_{3}+
x_{6}+ 4}++ z^{x_{3}+5} +z^{ x_{4}+ x_{5}+ x_{6}+5}+ z^{x_{4}+
x_{5}+4}+ z^{x_{4}+ x_{6}+ 4}+ z^{x_{5}+ x_{6}+ 4}+ z^{x_{4}+5}+
z^{x_{5}+6}+ z^{x_{6}+6}+ z^{ x_{3}+ x_{5}+ x_{6}+5}+ z^{x_{3}+
x_{5}+ 4}+ z^{x_{3}+ x_{6}+ 4}+ z^{x_{3}+5}+ z^{x_{4}+ x_{5}+
x_{6}+5}+ z^{x_{4}+ x_{5}+ 4}+ z^{x_{4}+ x_{6}+ 4}+ z^{x_{4}+5}+
z^{x_{5}+ x_{6}+ 4}+ z^{x_{5}+6}+ z^{x_{6}+6}+ z^{x_{3}+x_{4}+
x_{5}+ 1}+ z^{x_{3}+x_{4}+ x_{6}+ 1}+ z^{x_{3}+x_{4}+2}+ z^{x_{3}+
x_{5}+ x_{6}+ 1}+ z^{x_{3}+ x_{5}+3}+z^{x_{3}+ x_{6}+3}+ z^{x_{3}+
1}+ z^{x_{4}+x_{5}+ x_{6}+
1}+ z^{x_{4}+ x_{5}+3}+ z^{x_{4}+ x_{6}+3}+ z^{x_{4}+1}+ z^{x_{5}+
x_{6}+4}+ z^{x_{5}+1}+ z^{x_{6}+1} + z^{4} )\).
since \(2\leq r\leq s\leq t\leq u\),
Compare the l.r.p in \(Q_{11}(A)\) and
the l.r.p in \(Q_{22}(B)\) we have
\(s = 3\) or \(t = 3\) or \(u =
3\).
\(\underline{\mathbf{Case\ 2.3.1}:}\)
\(s = 3\).
Then \(A\cong\theta(3,3,3,3,t,u)\),
\(A\) is \(\chi\)-unique by Lemma 15.
\(\underline{\mathbf{Case\ 2.3.2}:}\)
\(t = 3\).
Since \(3\leq s\leq 3\leq u\). We know
that \(s = 3\).
Then \(A\cong\theta(3,3,3,3,3,u)\),
\(A\) is \(\chi\)-unique by Lemma 9.
\(\underline{\mathbf{Case\ 2.3.3}:}\)
\(u = 3\).
Since \(3\leq s\leq t\leq 3\). We know
that \(s = t = 3\).
Then \(A\cong\theta(3,3,3,3,3,3)\),
\(A\) is \(\chi\)-unique by Theorem 1.
\(\underline{\mathbf{Case\ C}:}\)
\(B \in g_{e} (\theta
(x_{1},x_{2},x_{3},x_{4},x_{5}),
C_{x_{6}+1})\), where \(2 \leq
x_{1}\leq x_{2} \leq
x_{3} \leq x_{4 }\leq x_{5}\) and \(2
\leq x_{6}\).
As \(A\) \(\cong \theta(r,r,s,s,t,u)\) and \(B\) \(\in g_{e}
(\theta (x_{1},x_{2},x_{3},x_{4},x_{5}),
C_{x_{6}+1})\), then by Theorem 7, we have
\(Q_{12}(A)=
z(z^{r}-1)^{2}(z^{s}-1)^{2}(z^{t}-1)(z^{u}-1)-
(z^{r}-z)^{3}(z^{s}-z)(z^{t}-z)(z^{u}-z)\).
\(Q_{23}(B) =
z(z^{x_{1}}-1)(z^{x_{2}}-1)(z^{x_{3}}-1)(z^{x_{4}}-1)(z^{x_{5}}-1)(z^{x_{6}}-1)-
(z^{x_{1}}-z)(z^{x_{2}}-z)(z^{x_{3}}-z)(z^{x_{4}}-z)(z^{x_{5}}-z)(z^{x_{6}}-1)\).
\(Q_{12}(A)= Q_{23}(B)\), yields
\(Q_{13}(A)=
z^{2r+2s+1}+z^{2r+t+u+1}+2z^{2r+s+u+1}+2z^{2r+s+t+1}+2z^{2r+u+t+1}+z^{2r+1}+z^{2s+1}+z^{t+u+1}+2z^{s+t+1}+2z^{s+u+1}
+2z^{r+s+1}+2z^{r+u+1}+2z^{r+2s+t+1}+2z^{r+2s+u+1}+4z^{r+s+t+u+1}+2z^{s+t+u+3}+2z^{r+t+u+3}+2z^{r+2s+t+u+1}+z^{2r+t+3}+z^{2s+t+3}
z^{t+5}+4z^{r+s+t+3}+z^{2r+u+3}+z^{2s+u+3}+z^{u+5}+4z^{r+s+u+3}+z^{r+s+3}+z^{r+2s+3}+2z^{s+5}+2z^{r+5}+4z^{r+s+1}
-(z^{2r+t+1}+z^{2r+s+1}+z^{2r+u+1}+z^{2s+t+1}+z^{2s+u+1}+z^{t+1}+2z^{s+1}+2z^{s+t+u+1}+2z^{r+1}+z^{u+1}+2z^{r+2s+t+u+1}+2z^{r+t+u+1}+4z^{r+s+t+1}
+4z^{r+s+u+1}+2z^{r+2s+1}+z^{2r+2s+2}+z^{2r+t+u+2}+z^{2r+4}+2z^{2r+s+t+2}+2z^{2r+s+u+2}+2z^{2s+t+u+2}+z^{2s+4}+z^{t+u+4}+2z^{s+t+4}
+2z^{s+u+4}+2z^{r+t+4}+2z^{r+u+4}+2z^{r+2s+t+2}+2z^{r+2s+u+2}+4z^{r+s+t+u+2}+4z^{r+s+4}+z^{6})\).
\(Q_{24}(B)= z^{x_{1}+ x_{2}+ x_{3}+ x_{4}+
x_{5}} + z^{x_{1}+ x_{2}+ x_{3}+ x_{6}+1}+ z^{x_{1}+ x_{2}+ x_{3}+2} +
z^{x_{1}+ x_{2}+ x_{4}+ x_{6}+1}
+ z^{x_{1}+ x_{2}+ x_{4}+2}+ z^{x_{1}+ x_{2}+ x_{5}+
x_{6}+1}+z^{x_{1}+ x_{2}+ x_{5}+2}+ z^{x_{1}+ x_{2}+ x_{6}+3}+
z^{x_{1}+ x_{2}+1} + z^{x_{1}+ x_{3}+ x_{4}+ x_{6}+1}+ z^{x_{1}+
x_{3}+ x_{4}+2}+ z^{x_{1}+ x_{3}+ x_{5}+ x_{6}+1}+ z^{x_{1}+ x_{3}+
x_{5}+ 2}+ z^{x_{1}+ x_{3}+ x_{6}+ 3}+ z^{x_{1}+ x_{3}+ 1}+
z^{x_{1}+ x_{4}+ x_{5}+ x_{6}+1}+ z^{x_{1}+ x_{4}+ x_{5}+ 2}+
z^{x_{1}+ x_{4}+ x_{6}+ 3}+ z^{x_{1}+ x_{4}+ 1}+ z^{x_{1}+ x_{5}+
x_{6}+ 3}+ z^{x_{1}+ x_{5}+1}+ z^{x_{1}+ x_{6}+1}+ z^{x_{1}+ 4}+
z^{x_{2}+ x_{3}+ x_{4}+ x_{6}+1}+ z^{x_{2}+ x_{3}+ x_{4}+2}+
z^{x_{2}+ x_{3}+ x_{5}+ x_{6}+1}+ z^{x_{2}+x_{3}+ x_{5}+ 2}+
z^{x_{2}+x_{3}+x_{6}+3}+z^{x_{2}+ x_{3}+ 1}+ z^{x_{2}+ x_{4}+ x_{5}+
x_{6}+1}+ z^{x_{2}+x_{4}+ x_{5}+ 2}+ z^{x_{2}+x_{4}+ x_{6}+ 3}+
z^{x_{2}+x_{4}+1}+ z^{x_{2}+ x_{5}+ x_{6}+ 3}+ z^{x_{2}+ x_{5}+1}+
z^{x_{2}+ x_{6}+1}+ z^{x_{2}+ 4}+ z^{x_{3}+ x_{4}+ x_{5}+ x_{6}+1}+
z^{x_{3}+x_{4}+ x_{5}+ 2}+ z^{x_{3}+x_{4}+ x_{6}+ 3}+
z^{x_{3}+x_{4}+1}+ z^{x_{3}+ x_{5}+ x_{6}+ 3}+
z^{x_{3}+ x_{5}+1}+z^{x_{3}+ x_{6}+1}+ z^{x_{3}+ 4}+
z^{x_{4}+x_{5}+ x_{6}+3}+ z^{x_{4}+ x_{5}+1}+ z^{x_{4}+
x_{6}+1}+ z^{x_{4}+4}+ z^{x_{5}+
x_{6}+1}+ z^{x_{5}+4}+ z^{x_{6}+5}-( z^{x_{1}+ x_{2}+ x_{3}+ x_{4}+
x_{5}+1} + z^{x_{1}+ x_{2}+ x_{3}+
x_{6}+2}+ z^{x_{1}+ x_{2}+ x_{3}+1}
+ z^{x_{1}+ x_{2}+ x_{4}+ x_{6}+2}+ z^{x_{1}+ x_{2}+ x_{4}+1} +
z^{x_{1}+ x_{2}+ x_{5}+
x_{6}+2}+z^{x_{1}+ x_{2}+ x_{5}+1} + z^{x_{1}+ x_{2}+ x_{6}+1}+
z^{x_{1}+x_{2}+3}+ z^{x_{1}+ x_{3}+ x_{4}+ x_{6}+1}+ z^{x_{1}+ x_{3}+
x_{4}+1}
+ z^{x_{1}+ x_{3}+ x_{5}+ x_{6}+2}+ z^{x_{1}+ x_{3}+ x_{5}+ 1}+
z^{x_{1}+ x_{3}+ x_{6}+ 1}++ z^{x_{1}+ x_{3}+3} +z^{x_{1}+ x_{4}+
x_{5}+ x_{6}+2}+ z^{x_{1}+ x_{4}+ x_{5}+ 1}+ z^{x_{1}+ x_{4}+
x_{6}+ 1}+ z^{x_{1}+ x_{5}+ x_{6}+ 1}+ z^{x_{1}+ x_{4}+3}+ z^{x_{1}+
x_{5}+3}+ z^{x_{1}+ x_{6}+4}+ z^{x_{1}+1}+z^{x_{2}+ x_{3}+ x_{4}+
x_{6}+2}+ z^{x_{2}+ x_{3}+ x_{4}+1}+ z^{x_{2}+ x_{3}+ x_{5}+
x_{6}+2}+ z^{x_{2}+x_{3}+ x_{5}+ 1}+ z^{x_{2}+x_{3}+ x_{6}+ 1}+
z^{x_{2}+x_{3}+3}+z^{x_{2}+ x_{4}+ x_{5}+ x_{6}+2}+ z^{x_{2}+ x_{4}+
x_{5}+1} +z^{x_{2}+x_{4}+ x_{5}+ x_{6}+2}+ z^{x_{2}+x_{4}+ x_{5}+
1}+ z^{x_{2}+x_{4}+ x_{6}+ 1}+ z^{x_{2}+x_{4}+3}+ z^{x_{2}+ x_{5}+
x_{6}+ 1}+ z^{x_{2}+ x_{5}+3}+ z^{x_{2}+ x_{6}+4}+ z^{x_{2}+ 1}+
z^{x_{3}+ x_{4}+ x_{5}+ x_{6}+2}+ z^{x_{3}+x_{4}+ x_{5}+ 1}+
z^{x_{3}+x_{4}+ x_{6}+ 1}+ z^{x_{3}+x_{4}+3}+ z^{x_{3}+ x_{5}+
x_{6}+ 1}+ z^{x_{3}+ x_{5}+3}+z^{x_{3}+ x_{6}+4}+ z^{x_{3}+ 1}+
z^{x_{4}+x_{5}+ x_{6}+ 1}+ z^{x_{4}+ x_{5}+3}+ z^{x_{4}+ x_{6}+4}+
z^{x_{4}+1}+ z^{x_{5}+ x_{6}+4}+ z^{x_{5}+1}+ z^{x_{6}+1} + z^{5}
)\).
Consider the l.r.p in \(Q_{13}(A)\)
that is r+1 and the l.r.p in \(Q_{24}(B)\) that is \(5\).
Since \(r = 4\) and r \(\geq\) \(2\). We have three cases to consider
1) \(r = 2\) or
2) \(r = 3\) or
3) \(r = 4\).
\(\underline{\mathbf{Case\ 1}:}\)
\(r = 2\).
Then \(g(A)= g(B)= 2r = 4\). Since
\(A\) has one cycle of length four,
therefore \(B\) has one cycle of length
\(4\). Without loss of generality, we
have two cases to consider,
1. \(x_{6}= 3\) or
2. \(x_{6} \neq 3\).
\(\underline{\mathbf{Case\ 1.1}:}\)
\(x_{6} = 3\).
Substituting \(r = 2\) in \(Q_{13}(A)\) and \(x_{6}= 3\) in \(Q_{24}(B)\). We obtain that there is \(-z^{3}\) in \(Q_{13}(A)\). Hence there are cases to be
considered, \(x_{1}= x_{2}= 2\) or
\(x_{1}=
x_{3}= 2\) or \(x_{1}= x_{4}=
2\) or \(x_{1}= x_{5}= 2\) or
\(x_{2}=
x_{3}= 2\) or \(x_{2}= x_{4}=
2\) or \(x_{2}= x_{5}= 2\) or
\(x_{3}=
x_{4}= 2\) or \(x_{3}= x_{5}=
2\) \(x_{4}= x_{5}= 2\). We know
that \(B\) has at least three cycles of
length \(4\) for all cases. Thus, a
contradiction.
\(\underline{\mathbf{Case\ 1.2}:}\)
\(x_{6} \neq 3\)
\(x_{6}\geq 4\).
Given that \(B\) has one cycle of
length \(4\), then \(x_{1}+ x_{2}= 4\), implying \(x_{1}= x_{2}= 2\). Consider the l.r.p in
\(Q_{13}(A)\) and the l.r.p in \(Q_{24}(B)\). We have \(s = 4\) or \(t =
4\) or \(u = 4\).
\(\underline{\mathbf{Case\ 1.2.1}:}\)
\(s = 4\). \[t + u + 8 = x_{3} + x_{4}+ x_{5}+
x_{6}.\tag{5}\]
\(Q_{14}(A)=
3z^{13}+3z^{t+u+5}+5z^{u+9}+5z^{t+9}+z^{t+u+1}+3z^{9}+2z^{t+5}+2z^{u+5}+3z^{u+11}
+3z^{t+11}+2z^{t+3}+2z^{u+3}+6z^{t+u+7}+6z^{7}-
(z^{t+1}+z^{u+1}+z^{5}+2z^{t+u+5}+2z^{t+u+3}+3z^{t+7}+3z^{u+7}+2z^{11}+z^{14}+z^{t+u+6}+2z^{t+10}+2z^{u+10}
+z^{12}+z^{t+u+4}+2z^{t+8}+2z^{u+8}+2z^{t+12}+2z^{u+12}+2z^{t+6}+2z^{u+6}+4z^{10}+z^{8}+z^{6})\).
\(Q_{25}(B)=
z^{x_{3}+x_{4}+x_{5}+4}+z^{x_{3}+x_{6}+5}+z^{x_{3}+6}+z^{x_{4}+x_{6}+5}+z^{x_{4}+6}+3z^{x_{5}+x_{6}+5}+z^{x_{5}+6}+z^{x_{6}+7}
+3z^{x_{3}+x_{4}+x_{6}+3}+2z^{x_{3}+x_{5}+4}+2z^{x_{3}+x_{6}+5}+2z^{x_{4}+3}+2z^{x_{5}+3}
+2z^{x_{3}+3}+3z^{x_{3}+x_{5}+x_{6}+3}+2z^{x_{3}+x_{5}+4}+2z^{6}+2z^{x_{3}+x_{6}+5}
+2z^{x_{6}+3}+3z^{x_{4}+x_{5}+x_{6}+3}+2z^{x_{4}+x_{5}+4}+2z^{x_{4}+x_{6}+5}+z^{x_{3}+x_{4}+x_{5}+2}+z^{x_{3}+x_{4}+1}+z^{x_{3}+x_{5}+1}
+z^{x_{3}+x_{6}+1}+z^{x_{4}+x_{5}+1}+z^{x_{4}+x_{6}+1}+z^{x_{5}+x_{6}+1}+z^{x_{4}+4}+z^{x_{5}+4}+z^{x_{6}+5}-(
z^{x_{3}+x_{4}+x_{5}+5}+z^{x_{3}+x_{6}+6}+z^{x_{3}+5}+z^{x_{4}+x_{6}+6}+z^{x_{4}+5}+z^{x_{5}+x_{6}+6}+3z^{x_{5}+5}+z^{x_{6}+5}
+z^{7}+z^{x_{3}+x_{4}+x_{6}+3}+3z^{x_{3}+x_{4}+3}
+2z^{x_{3}+x_{5}+x_{6}+4}+3z^{x_{3}+x_{5}+3}+2z^{x_{3}+x_{6}+3}+2z^{x_{3}+5}+2z^{x_{4}+x_{5}+x_{6}+4}
+3z^{x_{4}+x_{5}+3}+2z^{x_{4}+x_{6}+3}+2z^{x_{4}+5}+2z^{x_{5}+x_{6}+3}+2z^{x_{6}+6}+z^{x_{3}+x_{4}+x_{6}+4}
+z^{x_{3}+x_{4}+x_{5}+1}+z^{x_{3}+x_{4}+x_{6}+1}+z^{x_{3}+x_{5}+x_{6}+1}+z^{x_{3}+x_{6}+4}
+z^{x_{3}+1}+z^{x_{4}+x_{5}+x_{6}+1}+z^{x_{4}+x_{6}+4}+z^{x_{4}+1}+z^{x_{5}+x_{6}+4}+z^{x_{5}+1}+z^{x_{6}+1})\).
Compare the l.r.p in \(Q_{25}(A)\) and
the l.r.p in \(Q_{25}(B)\) we have
\(x_{3}= 4\) or \(x_{4}= 4\) or \(x_{5}= 4\) or \(x_{6}= 4\).
\(\underline{\mathbf{Case\ 1.2.1.1}:}\)
\(x_{3} = 4\).
\(Q_{26}(B)=
z^{x_{4}+x_{5}+8}+z^{x_{6}+9}+z^{10}+z^{x_{4}+x_{6}+5}+z^{x_{4}+6}+3z^{x_{5}+x_{6}+5}+z^{x_{5}+6}+z^{x_{6}+7}
+3z^{x_{4}+x_{6}+7}+2z^{x_{5}+8}+2z^{x_{6}+9}+2z^{x_{4}+3}+2z^{x_{5}+3}
+2z^{7}+3z^{x_{5}+x_{6}+7}+2z^{x_{5}+8}+2z^{6}+2z^{x_{6}+9}
+2z^{x_{6}+3}+3z^{x_{4}+x_{5}+x_{6}+3}+2z^{x_{4}+x_{5}+4}+2z^{x_{4}+x_{6}+5}+z^{x_{4}+x_{5}+6}+z^{x_{4}+5}+z^{x_{5}+5}
+z^{x_{6}+5}+z^{x_{4}+x_{5}+1}+z^{x_{4}+x_{6}+1}+z^{x_{5}+x_{6}+1}+z^{x_{4}+4}+z^{x_{5}+4}+z^{x_{6}+5}-(
z^{x_{4}+x_{5}+9}+z^{x_{6}+10}+z^{9}+z^{x_{4}+x_{6}+6}+z^{x_{4}+5}+z^{x_{5}+x_{6}+6}+3z^{x_{5}+5}+z^{x_{6}+5}
+z^{7}+z^{x_{4}+x_{6}+7}+3z^{x_{4}+7}
+2z^{x_{5}+x_{6}+8}+3z^{x_{5}+7}+2z^{x_{6}+7}+2z^{9}+2z^{x_{4}+x_{5}+x_{6}+4}
+3z^{x_{4}+x_{5}+3}+2z^{x_{4}+x_{6}+3}+2z^{x_{4}+5}+2z^{x_{5}+x_{6}+3}+2z^{x_{6}+6}+z^{x_{4}+x_{6}+8}
+z^{x_{4}+x_{5}+5}+z^{x_{4}+x_{6}+5}+z^{x_{5}+x_{6}+5}+z^{x_{6}+8}
+z^{x_{4}+x_{5}+x_{6}+1}+z^{x_{4}+x_{6}+4}+z^{x_{4}+1}+z^{x_{5}+x_{6}+4}+z^{x_{5}+1}+z^{x_{6}+1})\).
Compare the l.r.p in \(Q_{13}(A)\) and
the l.r.p in \(Q_{26}(B)\) we have
\(x_{4}=x_{5}=x_{6}= 5\). \[t + u = 11.\tag{6}\]
Since \(2\leq 4 \leq t\leq u\)
If \(t = 4\) then \(u = 7\), \(A\cong\theta(2,2,4,4,4,7)\), \(A\) is \(\chi\) unique by Lemma 10.
If \(t = 5\) then \(u = 6\).
We obtain \(Q_{13}(A)\neq Q_{26}(B)\),
a contradiction.
\(\underline{\mathbf{Case\ 1.2.1.2}:}\)
\(x_{4} = 4\). \[t + u + 4= x_{3}+ x_{5}+ x_{6}.\tag{7}\]
Similarly to Case 1.2.1.1, we obtain a contradiction.
\(\underline{\mathbf{Case\ 1.2.1.3}:}\)
\(x_{5} = 4\). \[t + u + 4= x_{3}+ x_{4}+ x_{6}.\tag{8}\]
Similarly to Case 1.2.1.1, we obtain a contradiction.
\(\underline{\mathbf{Case\ 1.2.1.4}:}\)
\(x_{6} = 4\). \[t + u + 4= x_{3}+ x_{5}+ x_{4}.\tag{9}\]
Similarly to Case 1.2.1.1, we obtain a contradiction.
\(\underline{\mathbf{Case\ 1.2.2}:}\)
\(t = 4\).
We know that either \(s =2\) or \(s = 3\) or \(s =
4\).
If \(s = 2\), \(A\cong\theta(2,2,2,2,4,u)\), \(A\) is \(\chi\)-unique by by Lemma 15.
If \(s = 3\).
\(Q_{15}(A)=
3z^{13}+6z^{u+8}+6z^{u+9}+6z^{12}+z^{u+11}+5z^{7}+2z^{u+5}+6z^{8}
+2z^{u+4}+4z^{11}+2z^{u+3}+6z^{u+10}+2z^{u+13}+z^{u+7}+3z^{6}-
(z^{u+5}+z^{u+1}+z^{8}+z^{11}+3z^{u+7}+3z^{u+8}+2z^{u+13}+5z^{10}+z^{12}+6z^{u+6}+6z^{9}+z^{u+10}
+z^{8}+2z^{u+9}+z^{u+12}+4z^{u+11}+2z^{13}+z^{8}+2z^{4}).\)
\(Q_{27}(B)=
z^{x_{3}+x_{4}+x_{5}+4}+z^{x_{3}+x_{6}+5}+z^{x_{3}+6}+z^{x_{4}+x_{6}+5}+z^{x_{4}+6}+3z^{x_{5}+x_{6}+5}+z^{x_{5}+6}+z^{x_{6}+7}
+3z^{x_{3}+x_{4}+x_{6}+3}+2z^{x_{3}+x_{5}+4}+2z^{x_{3}+x_{6}+5}+2z^{x_{4}+3}+2z^{x_{5}+3}
+2z^{x_{3}+3}+3z^{x_{3}+x_{5}+x_{6}+3}+2z^{x_{3}+x_{5}+4}+2z^{6}+2z^{x_{3}+x_{6}+5}
+2z^{x_{6}+3}+3z^{x_{4}+x_{5}+x_{6}+3}+2z^{x_{4}+x_{5}+4}+2z^{x_{4}+x_{6}+5}+z^{x_{3}+x_{4}+x_{5}+2}+z^{x_{3}+x_{4}+1}+z^{x_{3}+x_{5}+1}
+z^{x_{3}+x_{6}+1}+z^{x_{4}+x_{5}+1}+z^{x_{4}+x_{6}+1}+z^{x_{5}+x_{6}+1}+z^{x_{4}+4}+z^{x_{5}+4}+z^{x_{6}+5}-(
z^{x_{3}+x_{4}+x_{5}+5}+z^{x_{3}+x_{6}+6}+z^{x_{3}+5}+z^{x_{4}+x_{6}+6}+z^{x_{4}+5}+z^{x_{5}+x_{6}+6}+3z^{x_{5}+5}+z^{x_{6}+5}
+z^{7}+z^{x_{3}+x_{4}+x_{6}+3}+3z^{x_{3}+x_{4}+3}
+2z^{x_{3}+x_{5}+x_{6}+4}+3z^{x_{3}+x_{5}+3}+2z^{x_{3}+x_{6}+3}+2z^{x_{3}+5}+2z^{x_{4}+x_{5}+x_{6}+4}
+3z^{x_{4}+x_{5}+3}+2z^{x_{4}+x_{6}+3}+2z^{x_{4}+5}+2z^{x_{5}+x_{6}+3}+2z^{x_{6}+6}+z^{x_{3}+x_{4}+x_{6}+4}
+z^{x_{3}+x_{4}+x_{5}+1}+z^{x_{3}+x_{4}+x_{6}+1}+z^{x_{3}+x_{5}+x_{6}+1}+z^{x_{3}+x_{6}+4}
+z^{x_{3}+1}+z^{x_{4}+x_{5}+x_{6}+1}+z^{x_{4}+x_{6}+4}+z^{x_{4}+1}+z^{x_{5}+x_{6}+4}+z^{x_{5}+1}+z^{x_{6}+1}).\)
Compare the l.r.p in \(Q_{15}(A)\) and
the l.r.p in \(Q_{27}(B)\) we have
\(x_{3}=x_{4}=3\) or \(x_{3}=x_{5}=3\) or \(x_{4}=x_{5}=3\).
Without loss of generality, suppose that \(x_{3}=x_{4}=3\).
\(Q_{28}(B)=z^{x_{5}+10}+6z^{x_{6}+8}+2z^{9}+3z^{x_{5}+x_{6}+5}+z^{x_{5}+6}+z^{x_{6}+7}+3z^{x_{6}+9}+2z^{10}+6z^{x_{5}+x_{6}+6}
+4z^{x_{5}+7}+z^{x_{5}+8}+6z^{6}+2z^{x_{5}+3}+2z^{x_{6}+3}+z^{7}+2z^{x_{5}+4}+2z^{x_{6}+4}+z^{x_{5}+x_{6}+1}+z^{x_{5}+4}
+z^{x_{6}+4}+z^{7}
-(z^{x_{5}+11}+z^{x_{6}+9}+6z^{8}+z^{x_{6}+10}+z^{8}+z^{x_{5}+x_{6}+6}+3z^{x_{5}+5}+z^{x_{6}+5}+
z^{7}+z^{x_{6}+9}+3z^{9}+2z^{x_{5}+x_{6}+7}+3z^{x_{6}+6}
+2z^{x_{5}+6}+2z^{x_{5}+x_{6}+7}+3z^{x_{5}+6}+2z^{x_{6}+6}
+2z^{x_{5}+x_{6}+3}+2z^{x_{6}+6}+z^{x_{6}+10}+z^{x_{5}+7}+z^{x_{6}+7}
+z^{x_{5}+1}+3z^{x_{5}+x_{6}+4}+2z^{x_{6}+7}+z^{x_{6}+1}).\)
Compare the l.r.p in \(Q_{15}(A)\) and
\(Q_{28}(B)\), we have \(Q_{15}(A)\neq
Q_{27}(B)\), a contradiction.
If \(s=4\)
\(Q_{16}(A)=
z^{13}+z^{u+9}+2z^{u+9}+2z^{13}+2z^{u+9}+z^{5}+z^{9}+z^{u+5}+2z^{9}+2z^{u+5}
+2z^{7}+2z^{u+3}+2z^{15}+2z^{u+11}+4z^{u+11}+2z^{u+11}+2z^{u+9}+2z^{u+15}+z^{11}+z^{15}
z^{9}+4z^{13}+z^{u+7}+z^{u+11}+z^{u+5}+4z^{u+9}+z^{9}+z^{13}+2z^{9}+2z^{7}+4z^{7}
-(z^{9}+z^{9}+z^{u+5}+z^{13}+z^{u+9}+z^{5}+2z^{5}+2z^{u+9}+2z^{3}+z^{u+1}+2z^{u+15}+2z^{u+7}+4z^{11}
+4z^{u+7}+2z^{11}+z^{14}+z^{u+10}+z^{8}+2z^{14}+2z^{u+10}+2z^{u+14}+z^{12}+z^{u+8}+2z^{12}
+2z^{u+8}+2z^{10}+2z^{u+6}+2z^{16}+2z^{u+12}+4z^{u+12}+4z^{10}+z^{6})\).
\(Q_{29}(B)=
z^{x_{3}+x_{4}+x_{5}+4}+z^{x_{3}+x_{6}+5}+z^{x_{3}+6}+z^{x_{4}+x_{6}+5}+z^{x_{4}+6}+3z^{x_{5}+x_{6}+5}+z^{x_{5}+6}+z^{x_{6}+7}
+3z^{x_{3}+x_{4}+x_{6}+3}+2z^{x_{3}+x_{5}+4}+2z^{x_{3}+x_{6}+5}+2z^{x_{4}+3}+2z^{x_{5}+3}
+2z^{x_{3}+3}+3z^{x_{3}+x_{5}+x_{6}+3}+2z^{x_{3}+x_{5}+4}+2z^{6}+2z^{x_{3}+x_{6}+5}
+2z^{x_{6}+3}+3z^{x_{4}+x_{5}+x_{6}+3}+2z^{x_{4}+x_{5}+4}+2z^{x_{4}+x_{6}+5}+z^{x_{3}+x_{4}+x_{5}+2}+z^{x_{3}+x_{4}+1}+z^{x_{3}+x_{5}+1}
+z^{x_{3}+x_{6}+1}+z^{x_{4}+x_{5}+1}+z^{x_{4}+x_{6}+1}+z^{x_{5}+x_{6}+1}+z^{x_{4}+4}+z^{x_{5}+4}+z^{x_{6}+5}-(
z^{x_{3}+x_{4}+x_{5}+5}+z^{x_{3}+x_{6}+6}+z^{x_{3}+5}+z^{x_{4}+x_{6}+6}+z^{x_{4}+5}+z^{x_{5}+x_{6}+6}+3z^{x_{5}+5}+z^{x_{6}+5}
+z^{7}+z^{x_{3}+x_{4}+x_{6}+3}+3z^{x_{3}+x_{4}+3}
+2z^{x_{3}+x_{5}+x_{6}+4}+3z^{x_{3}+x_{5}+3}+2z^{x_{3}+x_{6}+3}+2z^{x_{3}+5}+2z^{x_{4}+x_{5}+x_{6}+4}
+3z^{x_{4}+x_{5}+3}+2z^{x_{4}+x_{6}+3}+2z^{x_{4}+5}+2z^{x_{5}+x_{6}+3}+2z^{x_{6}+6}+z^{x_{3}+x_{4}+x_{6}+4}
+z^{x_{3}+x_{4}+x_{5}+1}+z^{x_{3}+x_{4}+x_{6}+1}+z^{x_{3}+x_{5}+x_{6}+1}+z^{x_{3}+x_{6}+4}
+z^{x_{3}+1}+z^{x_{4}+x_{5}+x_{6}+1}+z^{x_{4}+x_{6}+4}+z^{x_{4}+1}+z^{x_{5}+x_{6}+4}+z^{x_{5}+1}+z^{x_{6}+1}).\)
Compare the l.r.p in \(Q_{16}(A)\) and
the l.r.p in \(Q_{29}(B)\) we have
\(x_{3}=x_{4}=2\) or \(x_{3}=x_{5}=2\) or \(x_{4}=x_{5}=2\).
Without loss of generality, suppose that \(x_{3}=x_{4}=3\).
\(Q_{30}(B)=
z^{x_{5}+8}+z^{x_{6}+7}+z^{8}+z^{x_{6}+7}+z^{8}+3z^{x_{5}+x_{6}+5}+z^{x_{5}+6}+z^{x_{6}+7}
+3z^{x_{6}+7}+2z^{x_{5}+6}+2z^{x_{6}+7}+2z^{5}+2z^{x_{5}+3}
+2z^{5}+3z^{x_{5}+x_{6}+5}+2z^{x_{5}+6}+2z^{6}+2z^{x_{6}+7}
+2z^{x_{6}+3}+3z^{x_{5}+x_{6}+5}+2z^{x_{5}+6}+2z^{x_{6}+7}+z^{x_{5}+6}+z^{5}+z^{x_{5}+3}
+z^{x_{6}+3}+z^{x_{5}+3}+z^{x_{6}+3}+z^{x_{5}+x_{6}+1}+z^{6}+z^{x_{5}+4}+z^{x_{6}+5}-(
z^{x_{5}+9}+z^{x_{6}+8}+z^{7}+z^{x_{6}+8}+z^{7}+z^{x_{5}+x_{6}+6}+3z^{x_{5}+5}+z^{x_{6}+5}
+z^{7}+z^{x_{6}+7}+3z^{7}
+2z^{x_{5}+x_{6}+6}+3z^{x_{5}+5}+2z^{x_{6}+5}+2z^{7}+2z^{x_{5}+x_{6}+6}
+3z^{x_{5}+5}+2z^{x_{6}+5}+2z^{7}+2z^{x_{5}+x_{6}+3}+2z^{x_{6}+6}+z^{x_{6}+8}
+z^{x_{5}+5}+z^{x_{6}+5}+z^{x_{5}+x_{6}+3}+z^{x_{6}+6}
+z^{3}+z^{x_{5}+x_{6}+3}+z^{x_{6}+6}+z^{3}+z^{x_{5}+x_{6}+4}+z^{x_{5}+1}+z^{x_{6}+1}).\)
Compare the l.r.p in \(Q_{16}(A)\) and
\(Q_{23}(B)\), we have \(Q_{16}(A)\neq
Q_{29}(B)\), a contradiction.
\(\underline{\mathbf{Case\ 1.2.3}:}\)
\(u = 4\).
Since \(2\leq s\leq t\leq 4\).
If \(s = 2\), then
\(t = 2\), \(A \cong\theta(2,2,2,2,2,4)\), \(A\) is \(\chi\)-unique by Lemma 9.
\(t = 3\), \(A \cong\theta(2,2,2,2,3,4)\), \(A\) is \(\chi\)-unique by Lemma 15.
\(t = 4\), \(A \cong\theta(2,2,2,2,4,4)\), \(A\) is \(\chi\)-unique by Lemma 9.
If \(s = 3\), then
\(t = 3\), \(A \cong\theta(2,2,3,3,3,4)\), \(A\) is \(\chi\)-unique by Lemma 14.
\(t = 4\), \(A \cong\theta(2,2,3,3,4,4)\), \(A\) is \(\chi\)-unique by Lemma 13.
If \(s = 4\), then
\(t = 4\), \(A \cong\theta(2,2,4,4,4,4)\), \(A\) is \(\chi\)-unique by Lemma 9.
\(\underline{\mathbf{Case\ 2}:}\)
\(r = 3\).
Then \(g(A)= g(B)= 2r = 6\). Since
\(A\) has a cycle of length \(6\), therefore \(B\) has one cycle of length \(6\). Without loss of generality, we have
two cases to consider ,
1. \(x_{6}= 5\) or
2. \(x_{6} \neq 5\).
\(\underline{\mathbf{Case\ 2.1}:}\)
\(x_{6}= 5\).
Substituting r = \(3\) in \(Q_{23}(A)\), we obtain that there is \(-2z^{4}\) in \(Q_{23}(A)\). There are cases to be
considered \(x_{1}=
x_{2}= 3\) or \(x_{1}= x_{3}=
3\) or \(x_{1}=x_{4}=3\) or
\(x_{1}=x_{5}=3\) or \(x_{2}= x_{3}=3\) or \(x_{2}= x_{4}=3\) or \(x_{2}= x_{5}=3\) or \(x_{3}= x_{4}=3\) or \(x_{3}= x_{5}=3\) or \(x_{4}=
x_{5}=3\).
We know that \(B\) has at least three
cycles of length \(6\) for all cases.
Thus a contradiction.
\(\underline{\mathbf{Case\ 2.2}:}\)
\(x_{6} \neq 5\).
Since the girth of \(B\) is \(6\), then \(x_{6}\geq 6\). Given that \(B\) has one cycle of length \(6\), then \(x_{1}+ x_{2}= 6\) implying \(x_{1}= x_{2}= 3\) .
\(Q_{17}(A)=z^{2s+7}+z^{t+u+7}+2z^{s+u+7}+2z^{t+u+1}+z^{2s+1}+2z^{s+t+1}+z^{7}+z^{t+u+1}+2z^{s+u+1}+2z^{2s+u+4}+2z^{2s+t+4}+2z^{t+4}
+2z^{u+4}+4z^{s+t+u+4}+2z^{s+t+u+3}+2z^{t+u+6}+z^{t+9}+z^{2s+t+3}+z^{t+5}+4z^{s+t+6}+z^{u+9}+z^{2s+u+3}+z^{u+5}+4z^{s+u+6}
+2z^{s+6}+2z^{s+5}+2z^{8}+2z^{2s+6}+4z^{s+4}-(3z^{t+7}+3z^{u+7}+5z^{s+7}+z^{2s+t+1}+z^{2s+u+1}+z^{t+1}+2z^{s+t+u+1}
+2z^{s+1}+4z^{t+u+4}+4z^{s+t+4}
+z^{2s+t+u+2}+2z^{2s+4}+z^{t+u+4}+z^{s+u+4}+z^{s+t+4}+z^{6}+2z^{2s+t+5}+2z^{2s+u+5}+4z^{s+t+u+5}+4z^{s+u+4}
+z^{u+1}+2z^{2s+4}+2z^{2s+8}+2z^{t+u+8}+z^{10}+2z^{s+u+8}+2z^{s+t+8})\).
\(Q_{31}(B)=z^{x_{3}+x_{4}+x_{5}+6}+z^{x_{3}+x_{6}+7}+z^{x_{3}+8}+z^{x_{4}+x_{6}+7}+z^{x_{4}+8}+z^{x_{5}+x_{6}+7}+z^{x_{5}+8}
+z^{x_{6}+8}+3z^{7}+2z^{x_{3}+x_{4}+x_{6}+4}+2z^{x_{3}+x_{4}+5}+2z^{x_{3}+x_{5}+x_{6}+4}
+2z^{x_{3}+x_{5}+5}+2z^{x_{3}+x_{6}+6}+3z^{x_{3}+4}+2z^{x_{4}+x_{5}+x_{6}+4}+2z^{x_{4}+x_{5}+5}
+2z^{x_{4}+x_{6}+6}+3z^{x_{4}+4}+2z^{x_{5}+x_{6}+6}+3z^{x_{5}+4}+2z^{x_{6}+4}+z^{x_{3}+x_{4}+x_{5}+x_{6}+1}
+z^{x_{3}+x_{4}+x_{5}+2}+z^{x_{3}+x_{4}+x_{6}+3}
+z^{x_{3}+x_{4}+1}+z^{x_{3}+x_{5}+1}+z^{x_{3}+x_{5}+x_{6}+3}+z^{x_{3}+x_{6}+1}+z^{x_{4}+x_{5}+x_{6}+3}
+z^{x_{4}+x_{5}+1}+z^{x_{4}+x_{6}+1}+z^{x_{6}+5}+z^{x_{5}+x_{6}+1}
-(z^{x_{3}+x_{4}+x_{5}+7}+z^{x_{3}+x_{6}+8}+z^{x_{3}+7}+z^{x_{4}+x_{6}+8}+z^{x_{4}+7}
+z^{x_{5}+x_{6}+8}+z^{x_{5}+7}+z^{x_{6}+7}+z^{9}+z^{x_{3}+x_{4}+x_{6}+4}
+2z^{x_{3}+x_{4}+4}+2z^{x_{3}+x_{5}+x_{6}+5}+2z^{x_{3}+x_{5}+4}+3z^{x_{3}+x_{6}+4}
+2z^{x_{3}+6}+2z^{x_{4}+x_{5}+x_{6}+5}+2z^{x_{4}+x_{5}+4}+3z^{x_{4}+x_{6}+4}
+2z^{x_{4}+6}+3z^{x_{5}+x_{6}+4}+2z^{x_{5}+6}+2z^{x_{6}+7}+2z^{x_{3}+x_{4}+x_{6}+5}+z^{x_{3}+x_{4}+x_{5}+1}+z^{x_{3}+x_{4}+x_{6}+1}
+z^{x_{3}+x_{4}+3}+z^{x_{3}+x_{5}+x_{6}+1}+z^{x_{3}+x_{5}+3}+z^{x_{3}+1}
+z^{x_{4}+x_{5}+x_{6}+1}+z^{x_{4}+x_{5}+3}+z^{x_{4}+1}+z^{x_{5}+1}+z^{x_{6}+1}+z^{5})\).
Compare the l.r.p in \(Q_{17}(A)\) and
the l.r.p in \(Q_{31}(B)\), we have
\(s = 4\) or \(t = 4\) or \(u =
4\).
\(\underline{\mathbf{Case\ 2.2.1}:}\)
\(s = 4\).
\(Q_{18}(A)=z^{13}+3z^{t+11}+3z^{u+11}+3z^{t+u+1}+3z^{9}+3z^{t+5}+3z^{u+5}+z^{7}+2z^{t+4}+2z^{u+4}
+3z^{t+u+8}+2z^{t+u+7}+2z^{t+u+6}+z^{t+u+7}+4z^{t+10}+4z^{u+10}+z^{10}+6z^{8}+2z^{14}-(
3z^{t+7}+3z^{u+7}+5z^{11}+z^{t+1}+z^{u+1}+2z^{t+u+5}+3z^{t+u+4}+6z^{t+8}+6z^{u+8}+2z^{16}+z^{t+u+10}
+z^{12}+2z^{t+13}+2z^{u+13}+4z^{t+u+9}+z^{6}+2z^{5})\).
Compare the l.r.p in \(Q_{18}(A)\) and
the l.r.p in \(Q_{31}(B)\), we have
\(-2z^{5}\) in \(Q_{18}(A)\) and \(-z^{5}\) in \(Q_{31}(B)\), so \(x_{3}= 4\) or \(x_{4}= 4\) or \(x_{5}= 4\).
\(\underline{\mathbf{Case\ 2.2.1.1}:}\)
\(x_{3}= 4\).
\(Q_{32}(B)=z^{x_{4}+x_{5}+10}+z^{x_{6}+11}+z^{12}+z^{x_{4}+x_{6}+7}+z^{x_{4}+8}+z^{x_{5}+x_{6}+7}+z^{x_{5}+8}
+z^{x_{6}+8}+3z^{7}+2z^{x_{4}+x_{6}+8}+2z^{x_{4}+9}+2z^{x_{5}+x_{6}+8}+2z^{x_{5}+9}+2z^{x_{6}+10}+3z^{8}
+2z^{x_{4}+x_{5}+x_{6}+4}+2z^{x_{4}+x_{6}+5}+2z^{x_{4}+x_{5}+6}
+3z^{x_{4}+4}+2z^{x_{5}+x_{6}+6}+3z^{x_{5}+4}+2z^{x_{6}+4}
+z^{x_{4}+x_{5}+x_{6}+5}+z^{x_{4}+x_{5}+1}+z^{x_{4}+x_{6}+1}+z^{x_{5}+x_{6}+1}+z^{x_{6}+5}-(
z^{x_{4}+x_{5}+11}+z^{x_{6}+12}+z^{11}+z^{x_{4}+x_{6}+8}+z^{x_{4}+7}+z^{x_{5}+x_{6}+8}+z^{x_{5}+7}+z^{x_{6}+7}+z^{9}
+z^{x_{4}+x_{6}+8}+2z^{x_{4}+8}+2z^{x_{5}+x_{6}+9}+2z^{x_{5}+8}
+3z^{x_{6}+8}+2z^{10}+2z^{x_{4}+x_{5}+4}+3z^{x_{4}+x_{6}+4}+2z^{x_{4}+6}+3z^{x_{5}+x_{6}+4}+2z^{x_{5}+6}+2z^{x_{6}+7}
+2z^{x_{4}+x_{6}+9}+2z^{x_{4}+x_{5}+5}
+z^{x_{4}+x_{6}+5}+z^{x_{4}+7}+z^{x_{5}+x_{6}+5}+z^{x_{5}+7}+z^{x_{4}+x_{5}+3}+z^{x_{4}+1}+z^{x_{5}+1}+z^{x_{6}+1})\).
Compare the l.r.p in \(Q_{18}(A)\) and
the l.r.p in \(Q_{32}(B)\), we have
\(x_{4} = 5\) or \(x_{5} = 5\).
\(\underline{\mathbf{Case\ 2.2.1.1.1}:}\)
\(x_{4} = 5\).
\(Q_{33}(B)=z^{x_{5}+15}+3z^{x_{6}+11}+z^{x_{6}+12}+z^{x_{5}+x_{6}+7}+z^{x_{5}+8}+z^{x_{6}+8}+2z^{7}+2z^{x_{5}+x_{6}+8}+z^{x_{5}+10}
+z^{x_{6}+10}+2z^{x_{5}+x_{6}+6}+3z^{x_{5}+4}+2z^{x_{6}+4}+z^{x_{5}+x_{6}+10}+z^{x_{6}+6}+z^{x_{5}+x_{6}+1}+z^{x_{6}+5}-(
z^{x_{5}+16}+z^{x_{6}+12}+3z^{11}+2z^{x_{5}+7}+3z^{x_{6}+7}+z^{9}+2z^{13}+3z^{x_{5}+8}+3z^{x_{6}+8}+2z^{10}+3z^{x_{6}+9}+z^{x_{6}+1}
+3z^{x_{5}+x_{6}+4}+z^{x_{5}+6}+2z^{x_{6}+14}+z^{x_{5}+x_{6}+5}+z^{x_{5}+1})\).
Compare the l.r.p in \(Q_{18}(A)\) and
the l.r.p in \(Q_{33}(B)\), we have
\(x_{5} = x_{6} = 6\).
\(Q_{19}(A)=3z^{t+11}+3z^{u+11}+3z^{t+u+1}+3z^{t+5}+3z^{u+5}+z^{7}+2z^{t+4}+2z^{u+4}
+3z^{t+u+8}+2z^{t+u+7}+2z^{t+u+6}+4z^{t+10}+4z^{u+10}+z^{10}+6z^{8}-(
3z^{t+7}+3z^{u+7}+5z^{11}+z^{t+1}+z^{u+1}+z^{t+u+5}+3z^{t+u+4}+6z^{t+8}+6z^{u+8}+2z^{16}+z^{t+u+10}
+2z^{t+13}+2z^{u+13}+2z^{t+u+9})\).
\(Q_{34}(B)=z^{21}+z^{19}+2z^{18}+2z^{17}+z^{16}+z^{12}+3z^{10}-(3z^{15}+4z^{14}+6z^{13}+2z^{11}+z^{9})\).
Compare the l.r.p in \(Q_{19}(A)\) and
the l.r.p in \(Q_{34}(B)\), we obtain
\(Q_{18}(A) \neq Q_{32}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 2.2.1.1.2}:}\)
\(x_{5} = 5\).
Similar to Case 2.2.1.1.1, we obtain a contradiction.
\(\underline{\mathbf{Case\ 2.2.1.2.1}:}\)
\(x_{4}= 4\).
Similar to Case 2.2.1.1.1, we obtain a contradiction.
\(\underline{\mathbf{Case\ 2.2.1.2.2}:}\)
\(x_{5}= 4\).
Similar to Case 2.2.1.1, we obtain a contradiction.
\(\underline{\mathbf{Case\ 2.2.2}:}\)
\(t = 4\).
Since \(3\leq s\leq 4\leq u\). We know
that \(s = 3\) or \(s = 4\).
If \(s = 3\), \(A\cong\theta(3,3,3,3,4,u)\), \(A\) is \(\chi\)-unique by Lemma 12.
If \(s = 4\), \(A\cong\theta(3,3,4,4,4,u)\), \(A\) is \(\chi\)-unique by Lemma 12.
\(\underline{\mathbf{Case\ 2.2.3}:}\)
\(u = 4\).
Since \(3\leq s\leq t\leq 4\). We know
that \(s = t = 3\) or \(s = 3\), \(t
=4\) or \(s = t =
4\).
If \(s = t = 3\), \(A\cong\theta(3,3,3,3,3,4)\), \(A\) is \(\chi\)-unique by Lemma 9.
If \(s = 3\), \(t =4\), \(A\cong\theta(3,3,3,3,4,4)\), \(A\) is \(\chi\)-unique by Lemma 9.
If \(s = t = 4\), \(A\cong\theta(3,3,4,4,4,4)\), \(A\) is \(\chi\)-unique by Lemma 9.
\(\underline{\mathbf{Case\ 3}:}\)
\(r = 4\).
Then \(g(A)= g(B)= 2r = 8\). Since
\(A\) has a cycle of length \(8\), therefore \(B\) has one cycle of length \(8\). Without loss of generality, we have
two cases to consider ,
1. \(x_{6}= 7\) or
2. \(x_{6} \neq 7\).
\(\underline{\mathbf{Case\ 3.1}:}\)
\(x_{6}= 7\).
Since \(B\) has one cycle of length
\(8\).
\(Q_{20}(A)=z^{2s+9}+z^{t+u+9}+3z^{9}+2z^{s+u+9}+2z^{s+t+9}+z^{t+u+1}+z^{2s+1}+2z^{s+t+1}+2z^{s+u+1}
+2z^{2s+t+5}+2z^{2s+u+5}+3z^{t+5}+3z^{u+5}+4z^{s+t+u+5}+z^{2s+t+u+1}
+2z^{s+t+u+3}+2z^{t+u+7}+z^{t+11}+z^{2s+t+3}+4z^{s+t+7}+z^{u+11}+z^{2s+u+3}+4z^{s+u+7}+2z^{s+7}+6z^{s+5}+2z^{2s+7}
-(z^{t+9}+z^{u+9}+2z^{s+9}+z^{2s+t+1}+z^{2s+u+1}+z^{t+1}+2z^{s+t+u+1}+2z^{s+1}+z^{6}+2z^{t+u+5}+z^{5}
+4z^{s+t+5}+4z^{s+u+5}+z^{u+1}+2z^{2s+5}+z^{2s+10}+z^{t+u+10}
+z^{12}+2z^{s+t+10}+2z^{s+u+10}+z^{2s+4}+z^{t+u+4}+2z^{s+u+4}+2z^{s+t+4}+2z^{2s+t+6}+2z^{2s+u+6}+2z^{t+8}+z^{2s+t+u+2}
+2z^{u+8}+4z^{s+t+u+6}+4z^{s+8})\).
\(Q_{34}(B)= z^{x_{1}+ x_{2}+ x_{3}+8}+
z^{x_{1}+ x_{2}+ x_{3}+2} + z^{x_{1}+ x_{2}+ x_{4}+ 8}
+ z^{x_{1}+ x_{2}+ x_{4}+2}+ z^{x_{1}+ x_{2}+ x_{5}+8}+z^{x_{1}+
x_{2}+ x_{5}+2}+ z^{x_{1}+ x_{2}+10}+ z^{x_{1}+ x_{2}+1} + z^{x_{1}+
x_{3}+ x_{4}+8}+ z^{x_{1}+ x_{3}+ x_{4}+2}+
z^{x_{1}+ x_{3}+ x_{5}+8}+ z^{x_{1}+ x_{3}+ x_{5}+ 2}+ z^{x_{1}+
x_{3}+10}+ z^{x_{1}+
x_{3}+ 1}+ z^{x_{1}+ x_{4}+ x_{5}+8}+ z^{x_{1}+ x_{4}+ x_{5}+ 2}+
z^{x_{1}+ x_{4}+10}+ z^{x_{1}+ x_{4}+ 1}+ z^{x_{1}+ x_{5}+10}+
z^{x_{1}+ x_{5}+1}+ z^{x_{1}+8}+ z^{x_{1}+ 4}+ + z^{x_{2}+ x_{3}+
x_{4}+8}+ z^{x_{2}+ x_{3}+ x_{4}+2}+ z^{x_{2}+ x_{3}+ x_{5}+8}+
z^{x_{2}+x_{3}+ x_{5}+ 2}+ z^{x_{2}+x_{3}+10}+z^{x_{2}+ x_{3}+1}+
z^{x_{2}+ x_{4}+ x_{5}+8}+ z^{x_{2}+x_{4}+ x_{5}+ 2}+
z^{x_{2}+x_{4}+10}+ z^{x_{2}+x_{4}+1}+ z^{x_{2}+ x_{5}+10}+
z^{x_{2}+ x_{5}+1}+ z^{x_{2}+8}+ z^{x_{2}+ 4}+ z^{x_{3}+ x_{4}+
x_{5}+ 8}+ z^{x_{3}+x_{4}+ x_{5}+ 2}+ z^{x_{3}+x_{4}+ 10}+
z^{x_{3}+x_{4}+1}+ z^{x_{3}+ x_{5}+ 10}+
z^{x_{3}+ x_{5}+1}+z^{x_{3}+8}+ z^{x_{3}+ 4}+
z^{x_{4}+x_{5}+ 10}+ z^{x_{4}+ x_{5}+1}+ z^{x_{4}+8}+ z^{x_{4}+4}+
z^{x_{5}+8}+z^{x_{5}+4}+z^{12}-
( z^{x_{1}+ x_{2}+ x_{3}+9}+ z^{x_{1}+ x_{2}+ x_{3}+1}
+ z^{x_{1}+ x_{2}+ x_{4}+9}+ z^{x_{1}+ x_{2}+ x_{4}+1} + z^{x_{1}+
x_{2}+ x_{5}+9}+z^{x_{1}+ x_{2}+ x_{5}+1} + z^{x_{1}+ x_{2}+8}+
z^{x_{1}+x_{2}+3}+ + z^{x_{1}+ x_{3}+ x_{4}+8}+ z^{x_{1}+ x_{3}+
x_{4}+1}
+ z^{x_{1}+ x_{3}+ x_{5}+9}+ z^{x_{1}+ x_{3}+ x_{5}+ 1}+ z^{x_{1}+
x_{3}+ 8}++ z^{x_{1}+ x_{3}+3} +z^{x_{1}+ x_{4}+ x_{5}+9}+
z^{x_{1}+x_{4}+ x_{5}+ 1}+ z^{x_{1}+ x_{4}+8}+ z^{x_{1}+ x_{5}+ 8}+
z^{x_{1}+ x_{4}+3}+ z^{x_{1}+ x_{5}+3}+ z^{x_{1}+11}+ z^{x_{1}+1}+
z^{x_{2}+ x_{3}+ x_{4}+9}+ z^{x_{2}+ x_{3}+ x_{4}+1}+ z^{x_{2}+
x_{3}+ x_{5}+ 9}+ z^{x_{2}+x_{3}+ x_{5}+ 1}+ z^{x_{2}+x_{3}+8}+
z^{x_{2}+x_{3}+3}+ + z^{x_{2}+ x_{4}+ x_{5}+9}+ z^{x_{2}+ x_{4}+
x_{5}+1} +z^{x_{2}+x_{4}+ x_{5}+9}+ z^{x_{2}+x_{4}+ x_{5}+ 1}+
z^{x_{2}+x_{4}+8}+ z^{x_{2}+x_{4}+3}+ z^{x_{2}+ x_{5}+8}+ z^{x_{2}+
x_{5}+3}+ z^{x_{2}+ x_{6}+4}+ z^{x_{2}+ 1}+ z^{x_{3}+ x_{4}+
x_{5}+9}+ z^{x_{3}+x_{4}+ x_{5}+ 1}+ z^{x_{3}+x_{4}+8}+
z^{x_{3}+x_{4}+3}+ z^{x_{3}+ x_{5}+8}+ z^{x_{3}+
x_{5}+3}+z^{x_{3}+11}+ z^{x_{3}+ 1}+ z^{x_{4}+x_{5}+8}+ z^{x_{4}+
x_{5}+3}+ z^{x_{4}+11}+ z^{x_{4}+1}+ z^{x_{5}+11}+ z^{x_{5}+1}+z^{8}
)\).
Compare the l.r.p in \(Q_{20}(A)\) and
the l.r.p in \(Q_{34}(B)\), we have
\(x_{1}= 4\).
\(Q_{35}(B)= z^{ x_{2}+ x_{3}+12}+ z^{x_{2}+
x_{3}+6} + z^{x_{2}+ x_{4}+ 12}
+ z^{x_{2}+ x_{4}+6}+ z^{x_{2}+ x_{5}+12}+z^{x_{2}+ x_{5}+6}+
z^{x_{2}+14}+ z^{x_{2}+5} + z^{x_{3}+ x_{4}+12}+ z^{x_{3}+ x_{4}+6}+
z^{x_{3}+ x_{5}+12}+ z^{x_{3}+ x_{5}+ 6}+ z^{x_{3}+14}+ z^{
x_{3}+ 5}+ z^{x_{4}+ x_{5}+12}+ z^{x_{4}+ x_{5}+ 6}+ z^{ x_{4}+14}+
z^{x_{4}+ 5}+ z^{x_{5}+14}+ z^{x_{5}+4}+ z^{12} + z^{x_{2}+ x_{3}+
x_{4}+8}+ z^{x_{2}+ x_{3}+ x_{4}+2}+ z^{x_{2}+ x_{3}+ x_{5}+8}+
z^{x_{2}+x_{3}+ x_{5}+ 2}+ z^{x_{2}+x_{3}+10}+z^{x_{2}+ x_{3}+1}+
z^{x_{2}+ x_{4}+ x_{5}+8}+ z^{x_{2}+x_{4}+ x_{5}+ 2}+
z^{x_{2}+x_{4}+10}+ z^{x_{2}+x_{4}+1}+ z^{x_{2}+ x_{5}+10}+
z^{x_{2}+ x_{5}+1}+ z^{x_{2}+8}+ z^{x_{2}+ 4}+ z^{x_{3}+ x_{4}+
x_{5}+ 8}+ z^{x_{3}+x_{4}+ x_{5}+ 2}+ z^{x_{3}+x_{4}+ 10}+
z^{x_{3}+x_{4}+1}+ z^{x_{3}+ x_{5}+ 10}+
z^{x_{3}+ x_{5}+1}+z^{x_{3}+8}+ z^{x_{3}+ 4}+
z^{x_{4}+x_{5}+ 10}+ z^{x_{4}+ x_{5}+1}+ z^{x_{4}+8}+ z^{x_{4}+4}+
z^{x_{5}+8}+z^{x_{5}+4}+2z^{12}-
( z^{x_{2}+ x_{3}+13}+ z^{x_{2}+ x_{3}+5}
+ z^{x_{2}+ x_{4}+13}+ z^{x_{2}+ x_{4}+5} + z^{x_{2}+
x_{5}+13}+z^{x_{2}+ x_{5}+5} + z^{x_{2}+12}
+ z^{x_{2}+7}+ + z^{x_{3}+ x_{4}+12}+ z^{x_{3}+ x_{4}+5} + z^{x_{3}+
x_{5}+13}+ z^{x_{3}+ x_{5}+5}+ z^{x_{3}+ 12}+
z^{x_{3}+7} +z^{x_{4}+ x_{5}+13}+ z^{x_{4}+ x_{5}+5}+ z^{ x_{4}+12}+
z^{x_{5}+ 12}+ z^{x_{4}+7}+ z^{x_{5}+7}+ z^{15} + z^{x_{2}+ x_{3}+
x_{4}+9}+ z^{x_{2}+ x_{3}+ x_{4}+1}+ z^{x_{2}+ x_{3}+ x_{5}+ 9}+
z^{x_{2}+x_{3}+ x_{5}+ 1}+ z^{x_{2}+x_{3}+8}+ z^{x_{2}+x_{3}+3}+ +
z^{x_{2}+ x_{4}+ x_{5}+9}+ z^{x_{2}+ x_{4}+ x_{5}+1}
+z^{x_{2}+x_{4}+ x_{5}+9}+ z^{x_{2}+x_{4}+ x_{5}+ 1}+
z^{x_{2}+x_{4}+8}+ z^{x_{2}+x_{4}+3}+ z^{x_{2}+ x_{5}+8}+ z^{x_{2}+
x_{5}+3}+ z^{x_{2}+ x_{6}+4}+ z^{x_{2}+ 1}+ z^{x_{3}+ x_{4}+
x_{5}+9}+ z^{x_{3}+x_{4}+ x_{5}+ 1}+ z^{x_{3}+x_{4}+8}+
z^{x_{3}+x_{4}+3}+ z^{x_{3}+ x_{5}+8}+ z^{x_{3}+
x_{5}+3}+z^{x_{3}+11}+ z^{x_{3}+ 1}+ z^{x_{4}+x_{5}+8}+ z^{x_{4}+
x_{5}+3}+ z^{x_{4}+11}+ z^{x_{4}+1}+ z^{x_{5}+11}+ z^{x_{5}+1}
)\).
Compare the l.r.p in \(Q_{20}(A)\) and
the l.r.p in \(Q_{35}(B)\), we have
\(x_{2}= 5\).
\(Q_{36}(B)= z^{ x_{3}+17}+ z^{x_{3}+11} +
z^{x_{4}+ 17}
+ z^{x_{4}+11}+ z^{x_{5}+17}+z^{x_{5}+11}+ z^{19}+ z^{10} +
z^{x_{3}+ x_{4}+12}+ z^{x_{3}+ x_{4}+6}+
z^{x_{3}+ x_{5}+12}+ z^{x_{3}+ x_{5}+ 6}+ z^{x_{3}+14}+ z^{
x_{3}+ 5}+ z^{x_{4}+ x_{5}+12}+ z^{x_{4}+ x_{5}+ 6}+ z^{ x_{4}+14}+
z^{x_{4}+ 5}+ z^{x_{5}+14}+ z^{x_{5}+4}+ z^{12} + z^{x_{3}+
x_{4}+13}+ z^{x_{3}+ x_{4}+7}+ z^{x_{3}+ x_{5}+13}+z^{x_{3}+ x_{5}+
7}+ z^{x_{3}+15}+z^{x_{3}+6}+ z^{ x_{4}+ x_{5}+13}+ z^{x_{4}+ x_{5}+
7}+ z^{x_{4}+15}+ z^{x_{4}+6}+ z^{x_{5}+15}+ z^{x_{5}+6}+ z^{13}+
z^{x_{3}+ x_{4}+ x_{5}+ 8}+ z^{x_{3}+x_{4}+ x_{5}+ 2}+
z^{x_{3}+x_{4}+ 10}+ z^{x_{3}+x_{4}+1}+ z^{x_{3}+ x_{5}+ 10}+
z^{x_{3}+ x_{5}+1}+z^{x_{3}+8}+ z^{x_{3}+ 4}+
z^{x_{4}+x_{5}+ 10}+ z^{x_{4}+ x_{5}+1}+ z^{x_{4}+8}+ z^{x_{4}+4}+
z^{x_{5}+8}+z^{x_{5}+4}+2z^{12}-
( z^{x_{3}+18}+ z^{x_{3}+10}
+ z^{x_{4}+18}+ z^{x_{4}+10}+ z^{x_{5}+18}+z^{x_{5}+10} + z^{17}
+ z^{12}+ + z^{x_{3}+ x_{4}+12}+ z^{x_{3}+ x_{4}+5} + z^{x_{3}+
x_{5}+13}+ z^{x_{3}+ x_{5}+5}+ z^{x_{3}+ 12}+
z^{x_{3}+7} +z^{x_{4}+ x_{5}+13}+ z^{x_{4}+ x_{5}+5}+ z^{ x_{4}+12}+
z^{x_{5}+ 12}+ z^{x_{4}+7}+ z^{x_{5}+7}+ z^{15} + z^{x_{2}+ x_{3}+
x_{4}+9}+ z^{x_{3}+ x_{4}+6}+ z^{x_{3}+ x_{5}+ 14}+ z^{x_{3}+ x_{5}+
6}+ z^{x_{3}+13}+ z^{x_{3}+8}+ + z^{x_{4}+ x_{5}+14}+ z^{ x_{4}+
x_{5}+6} +z^{x_{4}+ x_{5}+14}+ z^{x_{4}+ x_{5}+6}+ z^{x_{4}+13}+
z^{x_{4}+8}+ z^{x_{5}+13}+ z^{x_{5}+8}+ z^{ x_{6}+9}+ z^{x_{3}+
x_{4}+ x_{5}+9}+ z^{x_{3}+x_{4}+ x_{5}+ 1}+ z^{x_{3}+x_{4}+8}+
z^{x_{3}+x_{4}+3}+ z^{x_{3}+ x_{5}+8}+ z^{x_{3}+
x_{5}+3}+z^{x_{3}+11}+ z^{x_{3}+ 1}+ z^{x_{4}+x_{5}+8}+ z^{x_{4}+
x_{5}+3}+ z^{x_{4}+11}+ z^{x_{4}+1}+ z^{x_{5}+11}+ z^{x_{5}+1}
)\).
Compare the l.r.p in \(Q_{20}(A)\) and
the l.r.p in \(Q_{36}(B)\), we have
\(s = 5\) or \(s = 6\) or \(s =
7\) or \(s = 8\) or \(s \geq 9\).
\(\underline{\mathbf{Case\ 3.1.1}:}\)
\(s = 4\).
Since \(5 \leq x_{3}\leq x_{4}\leq
x_{5}.\) We have \(Q_{19}(A)\neq
Q_{34}(B)\).
\(\underline{\mathbf{Case\ 3.1.2}:}\)
\(s = 5\).
\(Q_{21}(A)=z^{19}+z^{t+u+9}+2z^{t+14}+2z^{u+14}+z^{t+u+1}+z^{11}+2z^{t+6}+2z^{u+6}+z^{u+13}
+3z^{t+5}+3z^{u+5}+3z^{t+u+10}+2z^{t+u+8}+4z^{u+12}+2z^{t+u+7}
+z^{t+13}+4z^{t+12}+2z^{12}+6z^{10}+2z^{17}+2z^{9}
-(3z^{t+9}+3z^{u+9}+3z^{14}+z^{t+1}+2z^{t+u+6}+2z^{6}+2z^{t+u+5}
+4z^{t+10}+4z^{u+10}+z^{u+1}+2z^{15}+z^{20}+z^{t+u+4}+2z^{t+16}+2z^{u+16}+2z^{t+8}+2z^{u+8}+4z^{t+u+11}+4z^{13})\).
\(Q_{37}(B)= z^{ x_{3}+17}+ z^{x_{3}+11} +
z^{x_{4}+ 17}
+ z^{x_{4}+11}+ z^{x_{5}+17}+z^{x_{5}+11}+ z^{19}+ z^{10} +
z^{x_{3}+ x_{4}+12}+ z^{x_{3}+ x_{4}+6}+
z^{x_{3}+ x_{5}+12}+ z^{x_{3}+ x_{5}+ 6}+ z^{x_{3}+14}+ z^{
x_{3}+ 5}+ z^{x_{4}+ x_{5}+12}+ z^{x_{4}+ x_{5}+ 6}+ z^{ x_{4}+14}+
z^{x_{4}+ 5}+ z^{x_{5}+14}+ z^{x_{5}+4}+ z^{12} + z^{x_{3}+
x_{4}+13}+ z^{x_{3}+ x_{4}+7}+ z^{x_{3}+ x_{5}+13}+z^{x_{3}+ x_{5}+
7}+ z^{x_{3}+15}+z^{x_{3}+6}+ z^{ x_{4}+ x_{5}+13}+ z^{x_{4}+ x_{5}+
7}+ z^{x_{4}+15}+ z^{x_{4}+6}+ z^{x_{5}+15}+ z^{x_{5}+6}+ z^{13}+
z^{x_{3}+ x_{4}+ x_{5}+ 8}+ z^{x_{3}+x_{4}+ x_{5}+ 2}+
z^{x_{3}+x_{4}+ 10}+ z^{x_{3}+x_{4}+1}+ z^{x_{3}+ x_{5}+ 10}+
z^{x_{3}+ x_{5}+1}+z^{x_{3}+8}+ z^{x_{3}+ 4}+
z^{x_{4}+x_{5}+ 10}+ z^{x_{4}+ x_{5}+1}+ z^{x_{4}+8}+ z^{x_{4}+4}+
z^{x_{5}+8}+z^{x_{5}+4}+2z^{12}-
( z^{x_{3}+18}+ z^{x_{3}+10}
+ z^{x_{4}+18}+ z^{x_{4}+10}+ z^{x_{5}+18}+z^{x_{5}+10} + z^{17}
+ z^{12}+ + z^{x_{3}+ x_{4}+12}+ z^{x_{3}+ x_{4}+5} + z^{x_{3}+
x_{5}+13}+ z^{x_{3}+ x_{5}+5}+ z^{x_{3}+ 12}+
z^{x_{3}+7} +z^{x_{4}+ x_{5}+13}+ z^{x_{4}+ x_{5}+5}+ z^{ x_{4}+12}+
z^{x_{5}+ 12}+ z^{x_{4}+7}+ z^{x_{5}+7}+ z^{15} + z^{x_{2}+ x_{3}+
x_{4}+9}+ z^{x_{3}+ x_{4}+6}+ z^{x_{3}+ x_{5}+ 14}+ z^{x_{3}+ x_{5}+
6}+ z^{x_{3}+13}+ z^{x_{3}+8}+ + z^{x_{4}+ x_{5}+14}+ z^{ x_{4}+
x_{5}+6} +z^{x_{4}+ x_{5}+14}+ z^{x_{4}+ x_{5}+6}+ z^{x_{4}+13}+
z^{x_{4}+8}+ z^{x_{5}+13}+ z^{x_{5}+8}+ z^{ x_{6}+9}+ z^{x_{3}+
x_{4}+ x_{5}+9}+ z^{x_{3}+x_{4}+ x_{5}+ 1}+ z^{x_{3}+x_{4}+8}+
z^{x_{3}+x_{4}+3}+ z^{x_{3}+ x_{5}+8}+ z^{x_{3}+
x_{5}+3}+z^{x_{3}+11}+ z^{x_{3}+ 1}+ z^{x_{4}+x_{5}+8}+ z^{x_{4}+
x_{5}+3}+ z^{x_{4}+11}+ z^{x_{4}+1}+ z^{x_{5}+11}+ z^{x_{5}+1}
)\).
Compare the l.r.p in \(Q_{21}(A)\) and
the l.r.p in \(Q_{37}(B)\), we have
\(x_{3}= x_{4}= 5\).
\(Q_{38}(B)=2z^{x_{5}+17}+z^{x_{5}+14}+2z^{x_{5}+12}+3z^{x_{5}+6}+z^{x_{5}+22}+z^{x_{5}+15}+z^{x_{5}+4}+2z^{22}+2z^{19}+3z^{20}+2z^{11}
-(3z^{x_{5}+10}+z^{x_{5}+13}+z^{x_{5}+11}+2z^{x_{5}+8}+z^{x_{5}+7}+z^{x_{5}+1}+z^{24}+z^{23}+3z^{18}+2z^{17}+z^{16}+2z^{15}+z^{12})\).
Compare the l.r.p in \(Q_{21}(A)\) and
the l.r.p in \(Q_{38}(B)\), we have
\(t = 5\) or \(t = 6\) or \(t =
7\) or \(t = 8\) or \(t \geq 9\).
\(\underline{\mathbf{Case\ 3.1.2.1}:}\)
\(t = 5\).
\(A \cong\theta( 4,4,5,5,5,u)\), \(A\) is \(\chi\)-unique by Lemma 1.
\(\underline{\mathbf{Case\ 3.1.2.2}:}\)
\(t = 6\), \(x_{5} = 6\).
\(Q_{22}(A)=
4z^{u+14}+2z^{u+6}+z^{u+13}+3z^{u+5}+2z^{u+12}+z^{u+7}+z^{u+16}+2z^{18}+z^{12}+z^{11}+2z^{17}+2z^{10}-
(3z^{u+9}+4z^{u+10}+z^{u+1}+2z^{u+8}+3z^{13}+z^{15}+2z^{22}+5z^{14}+z^{u+17})\).
\(Q_{39}(B)=z^{28}+z^{21}+2z^{22}+3z^{20}-(3z^{17}+z^{24}+3z^{18}+z^{12})\).
We have \(Q_{22}(A)\neq Q_{39}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.2.3}:}\)
\(t = 7\), \(x_{5} = 7\).
\(Q_{23}(A)=
4z^{u+14}+2z^{u+6}+z^{u+13}+3z^{u+5}+2z^{u+12}+2z^{u+15}+3z^{u+17}+z^{21}+4z^{12}+3z^{10}-
(3z^{u+9}+4z^{u+10}+z^{u+1}+z^{u+8}+4z^{13}+z^{16}+z^{23}+2z^{u+16}+2z^{u+16}+z^{u+18}+2z^{u+13}+2z^{14})\).
\(Q_{40}(B)=z^{24}+z^{13}+2z^{29}+3z^{22}+3z^{11}+z^{20}-(3z^{17}+2z^{15}+4z^{18}+z^{12})\).
We have \(Q_{23}(A)\neq Q_{40}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.2.4}:}\)
\(t = 8\), \(x_{5} = 8\).
\(Q_{24}(A)=
3z^{u+18}+2z^{u+6}+z^{u+13}+3z^{u+5}+4z^{u+12}+z^{21}+2z^{12}+3z^{10}-
(2z^{u+8}+z^{u+19}+z^{24}+z^{14}+z^{13}+2z^{u+9}+z^{u+1}+4z^{u+10})\).
\(Q_{41}(B)=2z^{25}+3z^{14}+z^{30}+z^{22}+z^{19}+2z^{20}+2z^{11}-
(z^{21}+z^{16}+3z^{15}+2z^{18}+z^{17}+z^{12})\).
We have \(Q_{24}(A)\neq Q_{41}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.2.5}:}\)
\(t \geq 9\), \(x_{5} \geq 9\)
\(Q_{25}(A)=2z^{u+14}+2z^{u+6}+z^{u+13}+3z^{u+5}+4z^{u+12}+z^{u+10}+3z^{u+19}+2z^{u+17}+2z^{23}+2z^{15}+3z^{10}+2z^{12}+4z^{21}-
(z^{u+20}+4z^{13}+3z^{u+9}+4z^{u+10}+z^{u+1}+2z^{u+16}+2z^{u+8}+2z^{25}+z^{19}+z^{20}+2z^{u+15})\).
\(Q_{42}(B)=
2z^{26}+2z^{21}+z^{15}+z^{31}+z^{13}+2z^{11}+2z^{19}+2z^{20}-
(4z^{17}+2z^{16}+z^{12})\).
We have \(Q_{25}(A)\neq Q_{42}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.3}:}\)
\(s = 6\).
\(Q_{26}(A)=z^{21}+3z^{t+u+9}+3z^{u+15}+3z^{t+15}+z^{t+u+1}+3z^{13}+3z^{u+13}+2z^{u+7}+2z^{t+7}
+2z^{t+17}+2z^{u+17}+3z^{t+5}+3z^{u+5}+4z^{t+u+11}+3z^{t+13}+6z^{11}+2z^{19}+2z^{9}
-(z^{t+9}+z^{u+9}+2z^{15}+z^{t+1}+2z^{7}+2z^{t+u+5}+3z^{t+11}+3z^{u+11}+z^{u+1}+2z^{17}+z^{22}
+z^{t+u+10}+2z^{t+16}+2z^{u+16}+z^{16}+z^{t+u+4}+2z^{t+10}+2z^{u+10}+2z^{t+18}+2z^{u+18}+2z^{t+8}+2z^{u+8}+4z^{t+u+12}+4z^{14})\).
\(Q_{43}(B)= z^{ x_{3}+17}+ z^{x_{3}+11} +
z^{x_{4}+ 17}
+ z^{x_{4}+11}+ z^{x_{5}+17}+z^{x_{5}+11}+ z^{19}+ z^{10} +
z^{x_{3}+ x_{4}+12}+ z^{x_{3}+ x_{4}+6}+
z^{x_{3}+ x_{5}+12}+ z^{x_{3}+ x_{5}+ 6}+ z^{x_{3}+14}+ z^{
x_{3}+ 5}+ z^{x_{4}+ x_{5}+12}+ z^{x_{4}+ x_{5}+ 6}+ z^{ x_{4}+14}+
z^{x_{4}+ 5}+ z^{x_{5}+14}+ z^{x_{5}+4}+ z^{12} + z^{x_{3}+
x_{4}+13}+ z^{x_{3}+ x_{4}+7}+ z^{x_{3}+ x_{5}+13}+z^{x_{3}+ x_{5}+
7}+ z^{x_{3}+15}+z^{x_{3}+6}+ z^{ x_{4}+ x_{5}+13}+ z^{x_{4}+ x_{5}+
7}+ z^{x_{4}+15}+ z^{x_{4}+6}+ z^{x_{5}+15}+ z^{x_{5}+6}+ z^{13}+
z^{x_{3}+ x_{4}+ x_{5}+ 8}+ z^{x_{3}+x_{4}+ x_{5}+ 2}+
z^{x_{3}+x_{4}+ 10}+ z^{x_{3}+x_{4}+1}+ z^{x_{3}+ x_{5}+ 10}+
z^{x_{3}+ x_{5}+1}+z^{x_{3}+8}+ z^{x_{3}+ 4}+
z^{x_{4}+x_{5}+ 10}+ z^{x_{4}+ x_{5}+1}+ z^{x_{4}+8}+ z^{x_{4}+4}+
z^{x_{5}+8}+z^{x_{5}+4}+2z^{12}-
( z^{x_{3}+18}+ z^{x_{3}+10}
+ z^{x_{4}+18}+ z^{x_{4}+10}+ z^{x_{5}+18}+z^{x_{5}+10} + z^{17}
+ z^{12}+ + z^{x_{3}+ x_{4}+12}+ z^{x_{3}+ x_{4}+5} + z^{x_{3}+
x_{5}+13}+ z^{x_{3}+ x_{5}+5}+ z^{x_{3}+ 12}+
z^{x_{3}+7} +z^{x_{4}+ x_{5}+13}+ z^{x_{4}+ x_{5}+5}+ z^{ x_{4}+12}+
z^{x_{5}+ 12}+ z^{x_{4}+7}+ z^{x_{5}+7}+ z^{15} + z^{x_{2}+ x_{3}+
x_{4}+9}+ z^{x_{3}+ x_{4}+6}+ z^{x_{3}+ x_{5}+ 14}+ z^{x_{3}+ x_{5}+
6}+ z^{x_{3}+13}+ z^{x_{3}+8}+ + z^{x_{4}+ x_{5}+14}+ z^{ x_{4}+
x_{5}+6} +z^{x_{4}+ x_{5}+14}+ z^{x_{4}+ x_{5}+6}+ z^{x_{4}+13}+
z^{x_{4}+8}+ z^{x_{5}+13}+ z^{x_{5}+8}+ z^{ x_{6}+9}+ z^{x_{3}+
x_{4}+ x_{5}+9}+ z^{x_{3}+x_{4}+ x_{5}+ 1}+ z^{x_{3}+x_{4}+8}+
z^{x_{3}+x_{4}+3}+ z^{x_{3}+ x_{5}+8}+ z^{x_{3}+
x_{5}+3}+z^{x_{3}+11}+ z^{x_{3}+ 1}+ z^{x_{4}+x_{5}+8}+ z^{x_{4}+
x_{5}+3}+ z^{x_{4}+11}+ z^{x_{4}+1}+ z^{x_{5}+11}+ z^{x_{5}+1}
).\)
Compare the l.r.p in \(Q_{26}(A)\) and
the l.r.p in \(Q_{43}(B)\), we have
\(x_{3}= x_{4}= 6\).
\(Q_{44}(B)=2z^{x_{5}+16}+z^{x_{5}+7}+z^{x_{5}+24}+z^{x_{5}+20}+z^{x_{5}+6}+z^{x_{5}+4}+z^{x_{5}+15}+2z^{23}+z^{20}+z^{25}+z^{21}+3z^{10}+3z^{12}+z^{22}
-(2z^{x_{5}+11}+2z^{x_{5}+9}+z^{x_{5}+14}+z^{x_{5}+21}+z^{x_{5}+10}+2z^{24}+z^{26}+2z^{19}+2z^{18}+2z^{16}+z^{x_{5}+1}).\)
Compare the l.r.p in \(Q_{26}(A)\) and
the l.r.p in \(Q_{44}(B)\), we have
\(t = 6\) or \(t = 7\) or \(t =
8\) or \(t \geq 9\).
\(\underline{\mathbf{Case\ 3.1.3.1}:}\)
\(t = 6\).
\(A \cong\theta( 4,4,6,6,6,u)\), \(A\) is \(\chi\)-unique by Lemma 14.
\(\underline{\mathbf{Case\ 3.1.
3.2}:}\) \(t = 7\), \(x_{5} = 7\).
\(Q_{27}(A)=z^{22}+3z^{u+15}+3z^{u+13}+2z^{14}+2z^{u+7}+2z^{24}+z^{u+8}+z^{u+17}+3z^{11}+3z^{u+5}+2z^{u+18}+2z^{20}+2z^{9}+2z^{13}
-(4z^{14}+2z^{15}+2z^{u+19}+z^{u+9}+2z^{u+12}+3z^{u+11}+z^{u+1}+2z^{u+10}+2z^{23}+2z^{25}+z^{17})\).
\(Q_{45}(B)=4z^{23}+z^{14}+z^{31}+z^{27}+3z^{10}+z^{25}-(3z^{16}+z^{28}+2z^{24}+z^{26}+2z^{19}+z^{18})\).
We have \(Q_{27}(A)\neq Q_{45}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.3.3}:}\)
\(t = 8\), \(x_{5} = 8\).
\(Q_{28}(A)=z^{25}+3z^{u+15}+z^{u+13}+z^{15}+2z^{u+7}+2z^{21}+4z^{11}+4z^{u+17}+6z^{13}+3z^{u+5}+4z^{u+19}+2z^{9}
–
(3z^{u+18}+2z^{u+20}+2z^{u+10}+2z^{u+8}+3z^{u+11}+z^{u+1}+4z^{14}+2z^{24}+2z^{u+16}+z^{17}+z^{26}+z^{22})\).
\(Q_{46}(B)=z^{23}+z^{28}+z^{14}+4z^{12}+3z^{10}+z^{20}-(z^{29}+z^{19}+z^{18}+2z^{18})\).
We have \(Q_{28}(A)\neq Q_{46}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.3.4}:}\)
\(t \geq 9\), \(x_{5} \geq 9\).
\(Q_{29}(A)=3z^{24}+3z^{u+15}+3z^{u+13}+2z^{u+7}+2z^{u+17}+3z^{u+5}+2z^{16}+z^{u+10}+2z^{26}+4z^{u+20}+z^{22}+2z^{9}+3z^{13}+4z^{11}-(
z^{u+9}+3z^{u+11}+z^{u+1}+z^{u+19}+2z^{u+16}+2z^{u+10}+2z^{u+8}+2z^{u+21}+2z^{u+14}+z^{14}+z^{20}+2z^{25}+2z^{27}+2z^{17}).\)
\(Q_{47}(B)=3z^{25}+z^{33}+z^{29}+z^{15}+z^{13}+3z^{12}+z^{23}+z^{20}+z^{21}+3z^{10}-
(5z^{18}+z^{30}+z^{26}+z^{24}+z^{19}+2z^{16})\).
We have \(Q_{29}(A)\neq Q_{47}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.4}:}\)
\(s = 7\).
\(Q_{30}(A)=z^{23}+3z^{t+u+9}+3z^{u+16}+3z^{t+16}+z^{t+u+1}+z^{15}+4z^{u+14}
+2z^{t+19}+2z^{u+19}+3z^{t+5}+3z^{u+5}+4z^{t+u+12}+z^{t+u+10}+2z^{t+u+7}+4z^{t+14}+6z^{12}+2z^{14}+2z^{9}+2z^{21}
-(z^{t+9}+z^{u+9}+2z^{16}+z^{t+1}+2z^{8}+2z^{t+u+8}+z^{u+15}+z^{t+15}+2z^{t+u+5}+4z^{t+12}+4z^{u+12}
+z^{u+1}+2z^{19}+z^{24}+z^{t+17}+z^{u+17}+z^{18}+z^{t+u+4}+2z^{t+20}+z^{t+11}+z^{u+11}+2z^{u+20}+4z^{t+u+13}+4z^{15}).\)
Compare the l.r.p in \(Q_{30}(A)\) and
the l.r.p in \(Q_{43}(B)\), we have
\(t = 7\) or \(t = 8\) or \(t
\geq 9\).
\(\underline{\mathbf{Case\ 3.1.4.1}:}\)
\(t = 7\).
\(A \cong\theta( 4,4,7,7,7,u)\), \(A\) is \(\chi\)-unique by Lemma 14.
\(\underline{\mathbf{Case\ 3.1.4.2}:}\)
\(Q_{31}(A)=z^{23}+3z^{u+16}+3z^{t+16}+z^{t+u+1}+z^{15}+4z^{u+14}
+2z^{t+19}+2z^{u+19}+3z^{t+5}+3z^{u+5}+4z^{t+u+12}+z^{t+u+10}+z^{t+u+7}+4z^{t+14}+6z^{12}+2z^{14}+2z^{9}
-(z^{t+9}+z^{u+9}+z^{16}+z^{t+1}+2z^{8}+2z^{t+u+8}+z^{u+15}+z^{t+15}+z^{t+u+5}+4z^{t+12}+4z^{u+12}
+z^{u+1}+z^{24}+z^{t+17}+z^{u+17}+z^{18}+2z^{t+20}+z^{t+11}+z^{u+11}+2z^{u+20}+2z^{t+u+13}+3z^{15}).\)
\(Q_{48}(B)=2z^{x_{5}+17}+2z^{x_{5}+19}+3z^{x_{5}+14}+z^{x_{5}+5}+z^{x_{5}+6}+z^{x_{5}+22}
+z^{x_{5}+26}+2z^{x_{5}+8}+z^{x_{5}+4}+3z^{24}
+z^{27}+z^{20}+z^{22}+3z^{13}+2z^{11}+z^{10}
-(z^{x_{}+18}+3z^{x_{5}+15}+z^{x_{5}+23}+z^{x_{5}+10}+z^{x_{5}+7}+z^{x_{5}+12}+2z^{25}+z^{28}+3z^{20}
+4z^{17}+2z^{14}+z^{x_{5}+1})\).
Compare the l.r.p in \(Q_{31}(A)\) is
\(2z^{9}\). We have \(t = u =
8\).
\(Q_{32}(A)=7z^{22}+6z^{13}+4z^{27}+z^{16}+z^{23}+z^{14}+6z^{12}-
(4z^{20}+2z^{25}+2z^{19}+2z^{29}+z^{21}+z^{17}+4z^{20}+3z^{15}+3z^{18}).\)
Compare the l.r.p in \(Q_{31}(B)\) is
\(z^{10}\). We have \(x_{5} =9\).
\(Q_{33}(A)=7z^{22}+6z^{13}+4z^{27}+z^{16}+z^{23}+z^{14}+6z^{12}-
(4z^{20}+2z^{25}+2z^{19}+2z^{29}+z^{21}+z^{17}+4z^{20}+3z^{15}+3z^{18})\).
\(Q_{49}(B)=2z^{26}+z^{28}+3z^{23}+z^{31}+z^{35}+4z^{13}+z^{22}+2z^{11}
-(3z^{24}+z^{32}+z^{19}+z^{16}+z^{21}+2z^{25}+2z^{20}+2z^{14})\).
We have \(Q_{33}(A)\neq Q_{49}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.4.3}:}\)
\(t \geq 9\), \(x_{5} \geq 9\).
\(Q_{34}(A)=3z^{u+16}+3z^{u+14}+3z^{u+5}+3z^{u+19}+z^{u+10}+4z^{u+21}+2z^{25}+2z^{23}+4z^{14}+2z^{28}+6z^{12}+2z^{9}
-(z^{u+9}+z^{u+15}+4z^{u+12}+z^{u+1}+z^{u+17}+z^{u+11}+2z^{u+20}+2z^{u+22}+2z^{u+17}+4z^{18}+2z^{24}+z^{10}+4z^{21}
+z^{26}+z^{20}+2z^{29}+3z^{15})\).
\(Q_{50}(B)=2z^{26}+z^{15}+z^{31}+z^{35}+z^{27}+4z^{13}+z^{22}+z^{27}+z^{22}
-(z^{32}+z^{19}+z^{16}+z^{21}+2z^{25}+2z^{20}+2z^{17}+2z^{14})\).
We have \(Q_{34}(A)\neq Q_{50}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.5}:}\)
\(s = 8\).
\(Q_{35}(A)=z^{25}+z^{t+u+9}+z^{t+17}+z^{u+17}+z^{t+u+1}+4z^{u+15}+z^{t+9}+z^{u+9}+2z^{t+21}+2z^{u+21}+3z^{t+5}
+3z^{u+5}+z^{u+11}+4z^{t+u+13}+2z^{t+u+11}+2z^{t+u+7}+z^{t+11}+z^{t+19}+4z^{t+15}+z^{u+19}+2z^{15}+6z^{13}+2z^{23}
-(z^{17}+z^{t+17}+z^{u+17}+z^{t+1}+2z^{t+u+9}+2z^{t+u+5}+4z^{t+13}+4z^{u+13}+z^{u+1}+2z^{21}+z^{26}+z^{t+u+10}
+2z^{t+8}+2z^{u+8}+z^{20}+z^{t+u+4}+2z^{t+12}+2z^{u+12}+2z^{t+22}+2z^{u+22}+2z^{t+8}+2z^{u+8}+4z^{t+u+14}+4z^{16})\).
We have \(t = 8\) or \(t \geq 9\).
\(\underline{\mathbf{Case\ 3.1.5.1}:}\)
\(t = 8\).
\(A \cong\theta( 4,4,8,8,8,u)\), \(A\) is \(\chi\)-unique by Lemma 14.
\(\underline{\mathbf{Case\ 3.1.5.2}:}\)
\(t \geq 9\).
\(Q_{36}(A)=z^{u+18}+z^{u+17}+z^{u+10}+4z^{u+15}+z^{u+9}+2z^{u+21}+3z^{u+5}+z^{u+11}+4z^{u+22}+2z^{u+20}+2z^{u+16}
+z^{25}+z^{18}+2z^{30}+3z^{14}+z^{20}+z^{28}+4z^{24}+2z^{15}+6z^{13}+2z^{23}
-(4z^{u+18}+2z^{u+14}+4z^{u+13}+z^{u+1}+2z^{u+12}+2z^{u+22}+2z^{u+8}+z^{u+19}+4z^{u+23}+3z^{17}+z^{u+13}+4z^{22}
+4z^{21}+2z^{27}+2z^{31}+4z^{16}+z^{10})\).
Consider the l.r.p in \(Q_{35}(A)\) is
\(-z^{10}\). We have \(x_{3}=9\).
\(Q_{51}(B)=z^{26}+z^{x_{4}+15}+z^{x_{4}+16}+z^{x_{5}+21}+z^{x_{5}+15}+z^{23}+z^{14}+z^{x_{4}+x_{5}+6}+z^{x_{4}+5}+z^{x_{5}+5}
+z^{x_{4}+22}+z^{24}+z^{x_{5}+16}+z^{x_{4}+6}+z^{x_{5}+6}+z^{x_{4}+x_{5}+17}+z^{x_{4}+x_{5}+21}+z^{x_{4}+19}
+z^{x_{5}+19}+2z^{13}+z^{x_{4}+4}+z^{x_{5}+4}+z^{19}+z^{12}+z^{10}
-(z^{x_{5}+18}+z^{x_{4}+18}+z^{19}+z^{x_{4}+12}+z^{x_{4}+7}+z^{x_{5}+12}+z^{x_{5}+7}+z^{x_{4}+23}+z^{x_{5}+23}+z^{x_{4}+13}+z^{x_{5}+13}
+z^{x_{4}+x_{5}+18}+z^{x_{4}+11}+z^{x_{5}+17}+z^{x_{5}+11}+z^{x_{4}+x_{5}+3}+z^{x_{4}+1}+z^{x_{5}+1}+z^{15}+z^{17})\).
Consider the l.r.p in \(Q_{36}(B)\) is
\(z^{10}\). We have \(x_{4}=9\).
\(Q_{37}(A)=z^{u+18}+z^{u+17}+z^{u+10}+3z^{u+15}+z^{u+9}+2z^{u+21}+3z^{u+5}+z^{u+11}+4z^{u+22}+z^{u+20}+2z^{u+16}
+z^{25}+z^{18}+2z^{30}+z^{14}+z^{20}+2z^{24}+3z^{13}+z^{23}
-(4z^{u+18}+z^{u+14}+4z^{u+13}+z^{u+1}+2z^{u+12}+z^{u+22}+2z^{u+8}+z^{u+19}+4z^{u+23}+2z^{17}+2z^{22}
+2z^{21}+2z^{31}+2z^{16})\).
\(Q_{52}(B)=z^{x_{5}+16}+z^{x_{5}+15}+z^{x_{5}+26}+z^{x_{5}+30}+z^{x_{5}+19}+z^{x_{5}+21}+z^{x_{5}+6}+z^{x_{5}+5}+z^{x_{5}+4}+z^{26}+z^{12}
-(z^{x_{5}+27}+z^{x_{5}+23}+z^{x_{5}+18}+z^{x_{5}+17}+z^{x_{5}+13}+2z^{x_{5}+12}+z^{x_{5}+11}+z^{x_{5}+7}+z^{x_{5}+1}+z^{32}+z^{20})\).
Consider the l.r.p in \(Q_{37}(B)\) is
\(z^{12}\). We have \(x_{5}=11\) \(u
\geq 9\).
\(Q_{53}(B)=2z^{26}+z^{16}+z^{17}+z^{37}+z^{41}+z^{30}+z^{15}+z^{31}
-(2z^{23}+z^{29}+z^{34}+z^{18}+z^{38}+z^{28}+z^{22}+z^{20}+z^{24})\).
We have \(Q_{37}(A)\neq Q_{53}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.1.6}:}\)
\(s \geq 9\).
\(Q_{38}(A)=z^{27}+z^{t+u+9}+2z^{u+18}+2z^{t+18}+z^{t+u+1}+z^{19}+4z^{u+16}+2z^{t+10}+2z^{u+10}+2z^{t+23}+2z^{u+23}
+3z^{t+5}+3z^{u+5}+z^{u+11}+4z^{t+u+14}+2z^{t+u+12}+2z^{t+u+7}+z^{t+11}+z^{t+21}+4z^{t+16}+z^{u+21}+2z^{16}+6z^{14}+2z^{25}
-(z^{t+9}+z^{u+9}+2z^{18}+z^{t+19}+z^{u+19}+z^{t+1}+2z^{t+u+10}+2z^{10}+2z^{t+u+5}+4z^{t+14}+4z^{u+14}+z^{u+1}
+2z^{23}+z^{28}+z^{t+u+10}+2z^{t+19}+2z^{u+19}+z^{22}+z^{t+u+4}+2z^{t+13}+2z^{u+13}+2z^{t+24}+2z^{u+24}+2z^{t+8}+2z^{u+8}+4z^{t+u+15}+4z^{17})\).
Consider the l.r.p in \(Q_{38}(A)\) is
\(2z^{9}\). We have \(x_{3}=x_{4}=5\).
\(Q_{54}(B)=2z^{22}+3z^{x_{5}+17}+z^{x_{5}+11}+z^{x_{5}+18}+z^{17}+3z^{19}+3z^{10}+z^{x_{5}+14}+z^{x_{5}+5}+z^{23}
+2z^{x_{5}+12}+3z^{20}+3z^{11}+3z^{x_{5}+15}+3z^{x_{5}+6}+z^{x_{5}+22}+z^{x_{5}+4}+z^{13}+z^{12}
-(2z^{23}+3z^{15}+3z^{x_{5}+10}+2z^{17}+2z^{12}+z^{x_{5}+12}+z^{x_{5}+7}+z^{24}+2z^{x_{5}+19}+z^{x_{5}+11}+2z^{18}+z^{13}
+2z^{x_{5}+13}+2z^{6}+z^{x_{5}+1}+z^{16}+z^{15}+z^{17})\).
Since \(9\leq t\leq u\), the the l.r.p
in \(Q_{54}(B)\) is\(-2z^{6}\) but not in \(Q_{38}(A)\).
We have \(Q_{38}(A)\neq Q_{54}(B)\), a
contradiction.
If \(x_{3}=x_{4}=9\).
We have \(Q_{38}(A)\neq Q_{54}(B)\), a
contradiction.
\(\underline{\mathbf{Case\ 3.2}:}\)
\(x_{6}\neq 7\).
We know that \(x_{6}>7\). Hence
\(x_{1} + x_{2}=8\) implies that \(x_{1} =
x_{2}=4\).
\(Q_{39}(A)=z^{2s+9}+z^{t+u+9}+3z^{9}+2z^{s+u+9}+2z^{s+t+9}+z^{t+u+1}+z^{2s+1}+2z^{s+t+1}+2z^{s+u+1}
+2z^{2s+t+5}+2z^{2s+u+5}+3z^{t+5}+3z^{u+5}+4z^{s+t+u+5}+z^{2s+t+u+1}
+2z^{s+t+u+3}+2z^{t+u+7}+z^{t+11}+z^{2s+t+3}+4z^{s+t+7}+z^{u+11}+z^{2s+u+3}+4z^{s+u+7}+2z^{s+7}+6z^{s+5}+2z^{2s+7}
-(z^{t+9}+z^{u+9}+2z^{s+9}+z^{2s+t+1}+z^{2s+u+1}+z^{t+1}+2z^{s+t+u+1}+2z^{s+1}+z^{6}+2z^{t+u+5}
+4z^{s+t+5}+4z^{s+u+5}+z^{u+1}+2z^{2s+5}+z^{2s+10}+z^{t+u+10}
+z^{12}+2z^{s+t+10}+2z^{s+u+10}+z^{2s+4}+z^{t+u+4}+2z^{s+u+4}+2z^{s+t+4}+2z^{2s+t+6}+2z^{2s+u+6}+2z^{t+8}
+2z^{u+8}+4z^{s+t+u+6}+4z^{s+8})\).
\(Q_{55}(B)= z^{x_{3}+ x_{4}+ x_{5}+8} +
z^{x_{3}+x_{6}+9}+ z^{x_{3}+10}
+ z^{x_{4}+ x_{6}+9} + z^{x_{4}+10}+ z^{x_{5}+
x_{6}+9}+z^{x_{5}+10}+ z^{x_{6}+11}+ z^{9} + 2z^{x_{3}+
x_{4}+x_{6}+5}+ 2z^{x_{3}+ x_{4}+6}+ 2z^{x_{3}+ x_{5}+ x_{6}+5}+
2z^{x_{3}+ x_{5}+6}+ 2z^{x_{3}+ x_{6}+7}+ 2z^{x_{3}+5}+ 2z^{x_{4}+
x_{5}+ x_{6}+5}+ 2z^{x_{4}+ x_{5}+ 6}+ 2z^{ x_{4}+ x_{6}+7}+ 2z^{
x_{4}+ 5}+ 2z^{x_{5}+ x_{6}+7}+ 2z^{x_{5}+5}+ 2z^{x_{6}+5}+ 2z^{8}+
z^{x_{3}+ x_{4}+ x_{5}+ x_{6}+1} + z^{x_{3}+x_{4}+ x_{5}+ 2}+
z^{x_{3}+x_{4}+ x_{6}+ 3}+ z^{x_{3}+x_{4}+1}+ z^{x_{3}+ x_{5}+
x_{6}+ 3}+
z^{x_{3}+ x_{5}+1}+z^{x_{3}+ x_{6}+1}+ z^{x_{3}+ 4}+
z^{x_{4}+x_{5}+ x_{6}+3}+ z^{x_{4}+ x_{5}+1}+ z^{x_{4}+
x_{6}+1}+ z^{x_{4}+4}+ z^{x_{5}+
x_{6}+1}+ z^{x_{5}+4}+ z^{x_{6}+5}-( z^{x_{3}+ x_{4}+ x_{5}+} + z^{
x_{3}+
x_{6}+10}+ z^{x_{3}+9}
+ z^{x_{4}+ x_{6}+10}+ z^{x_{4}+9} + z^{x_{5}+
x_{6}+10}+z^{x_{5}+9} + z^{x_{6}+9}+
z^{11}+z^{x_{3}+ x_{4}+ x_{6}+5}+z^{x_{3}+ x_{4}+ x_{6}+6}+ 2z^{x_{3}+
x_{4}+5}
+ 2z^{x_{3}+ x_{5}+ x_{6}+6}+ 2z^{x_{3}+ x_{5}+5}+ 2z^{x_{3}+ x_{6}+
5}+ 2z^{x_{3}+7}+2z^{x_{4}+ x_{5}+ x_{6}+6}+ 2z^{x_{4}+ x_{5}+5}+
2z^{x_{4}+ x_{6}+5}+ 2z^{x_{5}+ x_{6}+5}+ 2z^{x_{4}+7}+ 2z^{
x_{5}+7}+ z^{x_{6}+7}+z^{x_{6}+8}+ z^{x_{3}+x_{4}+ x_{5}+ 1}+
z^{x_{3}+x_{4}+ x_{6}+ 1}+ z^{x_{3}+x_{4}+3}+ z^{x_{3}+ x_{5}+
x_{6}+ 1}+ z^{x_{3}+ x_{5}+3}+z^{x_{3}+ x_{6}+4}+ z^{x_{3}+ 1}+
z^{x_{4}+x_{5}+ x_{6}+ 1}+ z^{x_{4}+ x_{5}+3}+ z^{x_{4}+ x_{6}+4}+
z^{x_{4}+1}+ z^{x_{5}+ x_{6}+4}+ z^{x_{5}+1}+ z^{x_{6}+1} + z^{5}
)\).
Compare the l.r.p in \(Q_{39}(A)\) and
the l.r.p in the \(Q_{55}(B)\). We have
\(s = 4\) or \(t = 4\) or \(u =
4\).
\(\underline{\mathbf{Case\ 3.2.1}:}\)
\(s = 4\)
\(A\cong\theta(4,4,4,4,t,u)\), \(A\) is \(\chi\)-unique by Lemma 15.
\(\underline{\mathbf{Case\ 3.2.2}:}\)
\(t = 4\).
We know that \(s = 4\).
\(A\cong\theta(4,4,4,4,4,u)\), \(A\) is \(\chi\)-unique by Lemma 9.
\(\underline{\mathbf{Case\ 3.2.3}:}\)
\(u = 4\).
We know that \(s = t = 4\).
\(A\cong\theta(4,4,4,4,4,4)\), \(A\) is \(\chi\)-unique by Theorem 1. ◻
Many researchers have studied the chromaticity of \(k\)-bridge graphs. A \(2\)-bridge graph is essentially a \(\chi\)-unique cycle graph. The theta graph, represented as \(\theta(1,y_1,y_2)\), serves as an example of a \(3\)-bridge graph. Chao and Whitehead [1] demonstrated that each such theta graph is \(\chi\)-unique. Extending this result, Loerinc [2] showed that all \(3\)-bridge graphs are \(\chi\)-unique. The chromaticity of \(4\)-bridge graphs was addressed by Chen et al. [3] and Xu et al. [4]. The chromaticity of \(5\)-bridge graphs has been the subject of investigation in several studies [5-8]. The chromaticity of the \(6\)-bridge graph has been explored by various researchers [11-19]. In this paper, we extend the existing research on the chromaticity of \(6\)-bridge graphs by investigating the chromaticity of a specific \(6\)-bridge graph, denoted as \(\theta(r,r,s,s,t,u)\), where \(2 \leq r \leq s \leq t \leq u\).
The authors are thankful to referee(s) for their useful comments and suggestions.