List packing number of bounded degree graphs

Combinatorics, Probability and Computing Pub Date : 2024-09-18 DOI:10.1017/s0963548324000191

Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, Ross J. Kang

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Abstract

We investigate the list packing number of a graph, the least

$k$

such that there are always

$k$

disjoint proper list-colourings whenever we have lists all of size

$k$

associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree

$\Delta$

has list packing number at most

$\Delta +1$

. Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks’-type theorem for the list packing number.

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有界阶数图的包装数列表

我们研究了图的列表包装数，即当我们有大小为 $k$ 的列表与顶点相关联时，总有 $k$ 不相交的适当列表着色的最小 $k$。我们很好奇列表包装数的行为与列表色度数的行为有什么不同，尤其是在有界度图的情况下。我们研究的主要问题是，是否每个最大度为 $\Delta$ 的图的列表打包数都最多为 $\Delta +1$ 。我们的结果凸显了列表打包的微妙之处，以及诸如寻求列表打包数布鲁克斯（Brooks'type）定理的障碍。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Combinatorics, Probability and Computing

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