Injective colorings of sparse graphs

TL;DR

This paper establishes bounds on the injective chromatic number of sparse graphs based on their maximum average degree and girth, showing that certain sparsity conditions guarantee optimal or near-optimal injective colorings.

Contribution

It provides new bounds linking maximum average degree and girth to the injective chromatic number, extending understanding of coloring properties in sparse graphs.

Findings

If mad(G) ≤ 5/2, then χ_i(G) ≤ Δ(G) + 1.

If mad(G) < 42/19, then χ_i(G) = Δ(G).

For planar graphs with girth ≥ 9, χ_i(G) ≤ Δ(G) + 1.

Abstract

Let $mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $\chi_i(G)$ denote the injective chromatic number of $G$. We prove that if $mad(G) \leq 5/2$, then $\chi_i(G)\leq\Delta(G) + 1$; and if $mad(G) < 42/19$, then $\chi_i(G)=\Delta(G)$. Suppose that $G$ is a planar graph with girth $g(G)$ and $\Delta(G)\geq 4$. We prove that if $g(G)\geq 9$, then $\chi_i(G)\leq\Delta(G)+1$; similarly, if $g(G)\geq 13$, then $\chi_i(G)=\Delta(G)$.