Injective colorings of graphs with low average degree

Let $\mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $χ_i(G)$ denote the injective chromatic number of $G$. We prove that if $Δ\geq 4$ and $\mad(G)<\frac{14}5$, then $χ_i(G)\leqΔ+2$. When $Δ=3$, we show that $\mad(G)<\frac{36}{13}$ implies $χ_i(G)\le 5$. In contrast, we give a graph $G$ with $Δ=3$, $\mad(G)=\frac{36}{13}$, and $χ_i(G)=6$.

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