Equitable list coloring of sparse graphs

TL;DR

This paper establishes sharp bounds for equitable list coloring of sparse graphs with minimum degree two, introducing the new concept of strongly equitable list coloring to achieve these results.

Contribution

It proves sharp bounds for equitable and equitable list coloring of specific sparse graphs, and introduces the new concept of strongly equitable list coloring.

Findings

Every (7/6,1/3)-sparse graph with minimum degree at least 2 is equitably 3-colorable and 3-choosable.

Every (5/4,1/2)-sparse graph with minimum degree at least 2 is equitably 4-colorable and 4-choosable.

The bounds are proven to be sharp.

Abstract

A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most $1$. For a list assignment $L$ of $k$ colors to each vertex of an $n$-vertex graph $G$, an equitable $L$-coloring of $G$ is a proper coloring of vertices of $G$ from their lists such that no color is used more than $\lceil n/k\rceil$ times. Call a graph equitably $k$-choosable if it has an equitable $L$-coloring for every $k$-list assignment $L$. A graph $G$ is $(a,b)$-sparse if for every $A\subseteq V(G)$, the number of edges in the subgraph $G[A]$ of $G$ induced by $A$ is at most $a|A|+b$. Our first main result is that every $(\frac{7}{6},\frac{1}{3})$-sparse graph with minimum degree at least $2$ is equitably $3$-colorable and equitably $3$-choosable. This is sharp. Our second main result is that every $(\frac{5}{4},\frac{1}{2})$-sparse graph with minimum degree at least $2$ is equitably $4$-colorable and equitably $4$-choosable. This is also sharp. One of the tools in the proof is the new notion of strongly equitable (SE) list coloring. This notion is both stronger and more natural than equitable list coloring; and our upper bounds are for SE list coloring.