AI 中文总结
本文针对不含三角形的平面图，证明了当接收3元列表的顶点集诱导稀疏匹配时，Hu和Zhu提出的列表着色猜想成立，为该猜想提供了新的关键佐证。
AI 中文摘要
若图G对每个k-列表分配都存在正常着色，则称G是k-可选择的。尽管所有不含三角形（$C_3$-free）的平面图都是4-可选择的，但Voigt构造出了其中一些并非3-可选择的例子。Hu和Zhu提出猜想：若G是不含三角形的平面图，且X⊆V(G)诱导出二分子图，则当X中每个顶点x的列表大小|L(x)|=3、V(G)\backslashX中每个顶点v的列表大小|L(v)|=4时，G存在正常L-着色。作为该猜想的佐证，他们证明了当X是独立集时猜想成立。本文进一步提供佐证，证明了当诱导子图G[X]是诱导稀疏匹配时猜想成立，这是首个支持该猜想的、接收较小列表的集合X可含边的结果。
英文摘要
A graph $G$ is $k$-choosable if it has a proper coloring for every $k$-list assignment. While every $C_3$-free planar graph is $4$-choosable, some of them are not $3$-choosable, as constructed by Voigt. Hu and Zhu conjectured that if $G$ is a $C_3$-free planar graph and $X \subseteq V(G)$ induces a bipartite subgraph, then $G$ has a proper $L$-coloring whenever $|L(x)| = 3$ for $x \in X$ and $|L(v)| = 4$ for $v \in V(G) \setminus X$. As evidence, they proved the conjecture when $X$ is an independent set. We provide further evidence by proving the conjecture when the induced subgraph $G[X]$ is an induced sparse matching. This is the first result supporting the conjecture in which the set $X$ receiving smaller lists may induce a subgraph with edges.
