2-distance list $(\Delta + 3)$-coloring of sparse graphs

TL;DR

This paper proves that sparse graphs with certain maximum average degree and maximum degree constraints can be properly colored with a 2-distance list coloring using at most Δ+3 colors.

Contribution

It establishes the existence of 2-distance list $( ext{max degree} + 3)$-colorings for specific classes of sparse graphs, extending previous coloring results.

Findings

Graphs with max average degree < 8/3 and Δ ≥ 4 are 2-distance list $( ext{Δ}+3)$-colorable.

Graphs with max average degree < 14/5 and Δ ≥ 6 are 2-distance list $( ext{Δ}+3)$-colorable.

The results apply to sparse graphs with bounded maximum average degree and degree.

Abstract

A 2-distance list k-coloring of a graph is a proper coloring of the vertices where each vertex has a list of at least k available colors and vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance list $(\Delta + 3)$-coloring for graphs with maximum average degree less than $\frac83$ and maximum degree $\Delta\geq 4$ as well as graphs with maximum average degree less than $\frac{14}5$ and maximum degree $\Delta\geq 6$.