arXiv 1302.25992013-02-11math.CO

不含相邻短圈的可平面图的$(3,1)^*$ -可选性

$(3,1)^*$-choosability of planar graphs without adjacent short cycles

Min Chen, Andre Raspaud

摘要：

图$G$的列表分配是将颜色的列表$L(v)$分配给每个顶点$v\in V(G)$的函数$L$。$(L,d)^*$ -着色是将颜色$π(v)\in L(v)$分配给每个顶点$v\in V(G)$的映射$π$，使得至多$v$的$d$邻居接收颜色$π(v)$。一个图$G$称为$(k,d)^*$ -可选的，如果它允许对每个列表分配$L$都有一个$(L,d)^*$ -着色，而对所有的$v\in V(G)$都有$|L(v)|\ge k$. 2001年Lih等在\cite{LSWZ-01}中证明了不含4-圈和$l$ -圈的可平面图是$(3,1)^*$ -可选的，其中$l\in \{5,6,7\}$ .后来，Dong和Xu证明了不含4-圈和l-圈的可平面图是$(3,1)^*$ -可选的，其中$l\in \{8,9\}$ .存在含有4-圈的可平面图，它们不是$(3,1)^*$ -可选择的（Crown，Crown and Woodall，1986 \cite{CCW-86}）.这部分地解释了在所有上述已知的可平面图的$(3,1)^*$ -可选性的充分条件中4-圈是完全禁止的。在本文中，我们允许4-圈不相邻相对较短的圈。更准确地说，我们证明了每个平面图没有4-圈相邻的3-和4-圈是$(3,1)^*$ -选择。这是对上述所有成果的共同加强。作为结果，我们还部分地回答了Xu和Zhang的一个问题\cite{XZ-07}，并证明了任何不含4-圈的平面图都是$(3,1)^*$ -可选的.

英文摘要：

A list assignment of a graph $G$ is a function $L$ that assigns a list $L(v)$ of colors to each vertex $v\in V(G)$. An $(L,d)^*$-coloring is a mapping $π$ that assigns a color $π(v)\in L(v)$ to each vertex $v\in V(G)$ so that at most $d$ neighbors of $v$ receive color $π(v)$. A graph $G$ is said to be $(k,d)^*$-choosable if it admits an $(L,d)^*$-coloring for every list assignment $L$ with $|L(v)|\ge k$ for all $v\in V(G)$. In 2001, Lih et al. \cite{LSWZ-01} proved that planar graphs without 4- and $l$-cycles are $(3,1)^*$-choosable, where $l\in \{5,6,7\}$. Later, Dong and Xu \cite{DX-09} proved that planar graphs without 4- and l-cycles are $(3,1)^*$-choosable, where $l\in \{8,9\}$. There exist planar graphs containing 4-cycles that are not $(3,1)^*$-choosable (Crown, Crown and Woodall, 1986 \cite{CCW-86}). This partly explains the fact that in all above known sufficient conditions for the $(3,1)^*$-choosability of planar graphs the 4-cycles are completely forbidden. In this paper we allow 4-cycles nonadjacent to relatively short cycles. More precisely, we prove that every planar graph without 4-cycles adjacent to 3- and 4-cycles is $(3,1)^*$-choosable. This is a common strengthening of all above mentioned results. Moreover as a consequence we give a partial answer to a question of Xu and Zhang \cite{XZ-07} and show that every planar graph without 4-cycles is $(3,1)^*$-choosable.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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