Linear Choosability of Sparse Graphs

TL;DR

This paper investigates the linear list chromatic number of sparse graphs, establishing exact values and bounds based on maximum average degree and girth, with implications for planar graphs.

Contribution

It provides new bounds and exact values for the linear list chromatic number of sparse graphs, including planar graphs with girth at least 5, advancing understanding of graph coloring.

Findings

For graphs with mad(G)<12/5 and Δ(G)≥3, lcl(G)=ceil(Δ(G)/2)+1.

For graphs with mad(G)<3 and Δ(G)≥9, lcl(G)≤ceil(Δ(G)/2)+2.

Planar graphs with girth≥5 have lcl(G)≤ceil(Δ(G)/2)+4.

Abstract

We study the linear list chromatic number, denoted $\lcl(G)$, of sparse graphs. The maximum average degree of a graph $G$, denoted $\mad(G)$, is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with maximum degree $\Delta(G)$ satisfies $\lcl(G)\ge \ceil{\Delta(G)/2}+1$. In this paper, we prove the following results: (1) if $\mad(G)<12/5$ and $\Delta(G)\ge 3$, then $\lcl(G)=\ceil{\Delta(G)/2}+1$, and we give an infinite family of examples to show that this result is best possible; (2) if $\mad(G)<3$ and $\Delta(G)\ge 9$, then $\lcl(G)\le\ceil{\Delta(G)/2}+2$, and we give an infinite family of examples to show that the bound on $\mad(G)$ cannot be increased in general; (3) if $G$ is planar and has girth at least 5, then $\lcl(G)\le\ceil{\Delta(G)/2}+4$.