arXiv 2107.081542021-07-17math.CO

DP色函数的一个删除-压缩关系

A Deletion-Contraction Relation for the DP Color Function

Jeffrey A. Mudrock

备注：: 16 pages

摘要：

DP-着色是列表着色的一种推广，由Dvoprák和Postle在2015年引入。图$G$的色多项式记为$P(G,m)$，等于$G$的正确$m$ -染色的个数.用于计算图$G$的色多项式的公知工具是将$P(G,m)$与两个较小图的色多项式联系起来的删除-收缩公式。图$G$的DP色函数，记为$P_{DP}(G,m)$，是色多项式的DP-染色模拟，而$P_{DP}(G,m)$是所有可能的$m$ -折叠覆盖上$G$的DP-染色的最小数目。本文给出了DP色函数的一个删除-压缩关系。为了使这成为可能，我们扩展的DP颜色函数的定义，多重图。我们还引入了图$G$的对偶DP色函数，记为$P^*_{DP}(G,m)$，它计算了$G$在某些$m$ -fold覆盖上的最大DP-着色数.我们展示了如何随着我们的删除-收缩关系的双DP色函数产生一个新的一般下界的DP色函数的图。

英文摘要：

DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvořák and Postle. The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. A well-known tool for computing the chromatic polynomial of graph $G$ is the deletion-contraction formula which relates $P(G,m)$ to the chromatic polynomials of two smaller graphs. The DP color function of a graph $G$, denoted $P_{DP}(G,m)$, is a DP-coloring analogue of the chromatic polynomial, and $P_{DP}(G,m)$ is the minimum number of DP-colorings of $G$ over all possible $m$-fold covers. In this paper we present a deletion-contraction relation for the DP color function. To make this possible, we extend the definition of the DP color function to multigraphs. We also introduce the dual DP color function of a graph $G$, denoted $P^*_{DP}(G,m)$, which counts the maximum number of DP-colorings of $G$ over certain $m$-fold covers. We show how the dual DP color function along with our deletion-contraction relation yields a new general lower bound on the DP color function of a graph.

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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