Fractional list packing for layered graphs

TL;DR

This paper investigates the fractional list packing number of graphs, providing bounds and improvements for related coloring concepts, especially for Cartesian products and d-degenerate graphs, with a focus on pathwidth.

Contribution

It introduces new bounds on the fractional list packing number, strengthening results for correspondence and local-degree versions, and relates it to pathwidth.

Findings

Bound on fractional list packing number by pathwidth plus one.

Improved theorems on flexible list coloring.

Showed the correspondence analogue fails for treewidth.

Abstract

The fractional list packing number $\chi_{\ell}^{\bullet}(G)$ of a graph $G$ is a graph invariant that has recently arisen from the study of disjoint list-colourings. It measures how large the lists of a list-assignment $L:V(G)\rightarrow 2^{\mathbb{N}}$ need to be to ensure the existence of a `perfectly balanced' probability distribution on proper $L$-colourings, i.e., such that at every vertex $v$, every colour appears with equal probability $1/|L(v)|$. In this work we give various bounds on $\chi_{\ell}^{\bullet}(G)$, which admit strengthenings for correspondence and local-degree versions. As a corollary, we improve theorems on the related notion of flexible list colouring. In particular we study Cartesian products and $d$-degenerate graphs, and we prove that $\chi_{\ell}^{\bullet}(G)$ is bounded from above by the pathwidth of $G$ plus one. The correspondence analogue of the latter is false for treewidth instead of pathwidth.