Injective colorings of graphs with low average degree

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 $\Delta \ge 4$ and $\mad(G)<\frac{14}5$, then ${\chi }_i(G)\le \Delta +2$. 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 ${\chi }_i(G)=6$.

15 pages, 3 figures