Packing list-colourings

TL;DR

This paper introduces and investigates the concept of list-packing in graph theory, focusing on the existence of multiple disjoint proper colourings from assigned lists, with results on bounds and special cases like bipartite graphs.

Contribution

It defines the list-packing problem as a new strengthening of list colouring and provides initial bounds and insights, especially for bipartite graphs, using combinatorial and probabilistic methods.

Findings

Established upper bounds on minimal k for list-packing

Analyzed list-packing in bipartite graphs

Connected list-packing to strong chromatic number

Abstract

List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a $k$-list-assignment $L$ of a graph $G$, which is the assignment of a list $L(v)$ of $k$ colours to each vertex $v\in V(G)$, we study the existence of $k$ pairwise-disjoint proper colourings of $G$ using colours from these lists. We may refer to this as a \emph{list-packing}. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest $k$ for which such a list-packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of $G$. (The reader might already find it interesting that such a minimal $k$ is well defined.) We also pursue a more focused study of the case when $G$ is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal $k$ above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalisations of the problem above in the same spirit.