Two smaller upper bounds of list injective chromatic number

Abstract

An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor receive distinct colors. Let $\chi _{i}^{l}(G)$ denote the list injective chromatic number of $G$. We prove that (1) $\chi _{i}^{l}(G)=\Delta$ for a graph $G$ with the maximum average degree $Mad(G)\le \frac{18}{7}$ and maximum degree $\Delta \ge 9$; (2) $\chi _{i}^{l}(G)\le \Delta +2$ if $G$ is a plane graph with $\Delta \ge 21$ and without 3-, 4-, 8-cycles.