AI 中文总结
研究4-不规则图平方的列表染色问题，通过证明得出对于此类图\(\chi_l(G^2) \leq 11\)，并给出例子说明此界是紧的，对克兰斯顿和金的相关研究进行了补充。
AI 中文摘要
图\(G\)的平方是通过在距离为\(2\)的任意两个顶点之间添加一条边而从\(G\)得到的图。\(k -\)不规则图是最大度为\(k\)且度为\(k\)的顶点不相邻的图。图的列表赋值是为每个顶点分配允许颜色列表的函数\(L\)。若存在恰当染色\(f\)使得对每个顶点\(v\)都有\(f(v) \in L(v)\)，则称该图是\(L -\)可染色的。若对于每个列表大小恰好为\(k\)的列表赋值，图\(G\)都是\(L -\)可染色的，则称图\(G\)是\(k -\)可选择的。图\(G\)的列表色数\(\chi_l(G)\)是使得\(G\)是\(k -\)可选择的最小整数\(k\)。克兰斯顿和金表明除彼得森图外所有次立方图都有\(\chi_l(G^2) \leq 8\)。他们还猜想对于最大度为\(k\)且最大团大小\(w (G^2)\leq k^2 - 1\)的图，有\(\chi_l(G^2) \leq k^2 - 1\)。我们证明对于\(4 -\)不规则图\(G\)，有\(\chi_l(G^2) \leq 11\)，并且给出一个例子表明这个界是紧的。
英文摘要
The square of a graph $G$ is the graph obtained from $G$ after adding an edge between any two vertices of distance $2$. A $k$-irregular graph is a graph with maximum degree $k$ such that vertices of degree $k$ are not adjacent. A \textit{list assignment} of a graph is a function $L$ that assigns to each vertex a list of permissible colors. The graph is said to be \textit{$L$-colorable} if there exists a proper coloring $f$ such that $f(v) \in L(v)$ for every vertex $v$. A graph $G$ is called \textit{$k$-choosable} if it is $L$-colorable for every list assignment where each list has exactly $k$ colors. The \textit{list chromatic number} of $G$, denoted by $χ_l(G)$, is the smallest integer $k$ for which $G$ is $k$-choosable. Cranston and Kim \cite{ck} showed that $χ_l(G^2) \leq 8$ for all subcubic graphs except the Petersen Graph. Moreover, Cranston and Kim \cite{ck} conjectured that for graphs with maximum degree $k$ and maximum clique size $w (G^2)\leq k^2-1$, we have $χ_l(G^2) \leq k^2-1$. We prove that for a 4-irregular graph $G$, we have $χ_l(G^2) \leq 11$. Moreover, we provide an example to show that this bound is sharp.
