既不含$K_5^{-}$也不含6-圈的Toroidal图是4-可选的
Toroidal graphs containing neither $K_5^{-}$ nor 6-cycles are 4-choosable
摘要：
图$G$的可选性$χ_\ell(G)$是最小$k$，使得在每个顶点处具有可用的$k$颜色保证适当的着色。给定一个环形图$G$，已知有$χ_\ell(G)\leq 7$和$χ_\ell(G)=7$当且仅当$G$包含$K_7$。Cai，Wang和Zhu证明了不含7-圈的环形图$G$是6-可选的，并且$χ_\ell(G)=6$当且仅当$G$包含$K_6$ .证明了不含6-圈的环形图$G$是5-可选的，并推测了$χ_\ell(G)=5$当且仅当$G$包含$K_5$ .我们通过构造一个无限族的非4-可着色的环面图来反驳这个猜想，其中既没有$K_5$，也没有长度至少为6的圈;而且，这个图族是可嵌入除了平面和射影平面之外的任何表面上的。相反，我们证明了以下稍弱的陈述建议朱：环形图既不包含$K^-_5$（一个$K_5$失踪的一个边缘），也不是6-圈是4-choosable。这是尖锐的，因为禁止两个结构中的一个并不能确保图是4-可选的。
英文摘要：
The choosability $χ_\ell(G)$ of a graph $G$ is the minimum $k$ such that having $k$ colors available at each vertex guarantees a proper coloring. Given a toroidal graph $G$, it is known that $χ_\ell(G)\leq 7$, and $χ_\ell(G)=7$ if and only if $G$ contains $K_7$. Cai, Wang, and Zhu proved that a toroidal graph $G$ without 7-cycles is 6-choosable, and $χ_\ell(G)=6$ if and only if $G$ contains $K_6$. They also prove that a toroidal graph $G$ without 6-cycles is 5-choosable, and conjecture that $χ_\ell(G)=5$ if and only if $G$ contains $K_5$. We disprove this conjecture by constructing an infinite family of non-4-colorable toroidal graphs with neither $K_5$ nor cycles of length at least 6; moreover, this family of graphs is embeddable on every surface except the plane and the projective plane. Instead, we prove the following slightly weaker statement suggested by Zhu: toroidal graphs containing neither $K^-_5$ (a $K_5$ missing one edge) nor 6-cycles are 4-choosable. This is sharp in the sense that forbidding only one of the two structures does not ensure that the graph is 4-choosable.
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