Injective colorings of graphs with low average degree

TL;DR

This paper establishes bounds on the injective chromatic number of graphs based on maximum degree and average degree, providing new thresholds for graphs with low average degree.

Contribution

It introduces new bounds on the injective chromatic number for graphs with certain maximum degree and average degree conditions, including tight examples.

Findings

For Δ ≥ 4, if mad(G) < 14/5, then χ_i(G) ≤ Δ + 2.

For Δ = 3, if mad(G) < 36/13, then χ_i(G) ≤ 5.

There exists a graph with Δ=3, mad(G)=36/13, and χ_i(G)=6.

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 $\Delta\geq 4$ and $\mad(G)<\frac{14}5$, then $\chi_i(G)\leq\Delta+2$. When $\Delta=3$, we show that $\mad(G)<\frac{36}{13}$ implies $\chi_i(G)\le 5$. In contrast, we give a graph $G$ with $\Delta=3$, $\mad(G)=\frac{36}{13}$, and $\chi_i(G)=6$.