List packing number of bounded degree graphs

TL;DR

This paper explores the list packing number in graphs, especially bounded degree graphs, revealing differences from the list chromatic number and examining whether it can be bounded by the maximum degree plus one.

Contribution

It investigates the properties of the list packing number in bounded degree graphs and discusses the challenges in establishing bounds similar to Brooks' theorem.

Findings

List packing number can differ significantly from list chromatic number.

Bounded degree graphs may not always have list packing number at most degree plus one.

Barriers exist to extending Brooks'-type bounds to list packing numbers.

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.