Bulletin of the
Korean Mathematical Society BKMS

A vertex coloring of a graph $G$ is called injective if any two vertices with a common neighbor receive distinct colors. A graph $G$ is injectively $k$-choosable if any list $L$ of admissible colors on $V(G)$ of size $k$ allows an injective coloring $\varphi$ such that $\varphi(v)\in L(v)$ whenever $v\in V(G)$. The least $k$ for which $G$ is injectively $k$-choosable is denoted by $\chi_{i}^{l}(G)$. For a planar graph $G$, Bu et al.~proved that $\chi_{i}^{l}(G)\leq\Delta+6$ if girth $g\geq5$ and maximum degree $\Delta(G)\geq8$. In this paper, we improve this result by showing that $\chi_{i}^{l}(G)\leq\Delta+6$ for $g\geq5$ and arbitrary $\Delta(G)$.

Keywords: Planar graph, list injective coloring, girth

MSC numbers: 05C15