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 $\Delta\geq 4$ 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 $\chi_i(G)=6$.Comment: 15 pages, 3 figure