Injective colorings of sparse graphs

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 $mad(G) \leq 5/2$, then $χ_i(G)\leqΔ(G) + 1$; and if $mad(G) < 42/19$, then $χ_i(G)=Δ(G)$. Suppose that $G$ is a planar graph with girth $g(G)$ and $Δ(G)\geq 4$. We prove that if $g(G)\geq 9$, then $χ_i(G)\leqΔ(G)+1$; similarly, if $g(G)\geq 13$, then $χ_i(G)=Δ(G)$.

No reviews yet.

No comments yet.