A smaller upper bound for the list injective chromatic number of planar graphs

An injective vertex coloring of a graph $G$ is a coloring where no two vertices that share a common neighbor are assigned the same color. If for any list $L$ of permissible colors with size $k$ assigned to the vertices $V(G)$ of a graph $G$, there exists an injective coloring $\phi$ in which $\phi (v)\in L(v)$ for each vertex $v\in V(G)$, then $G$ is said to be injectively $k$-choosable. The notation ${\chi }_i^l(G)$ represents the minimum value of $k$ such that a graph $G$ is injectively $k$-choosable. In this article, for any maximum degree $\mathrm{\Delta }$, we demonstrate that ${\chi }_i^l(G)\le \mathrm{\Delta }+4$ if $G$ is a planar graph with girth $g\ge 5$ and without intersecting 5-cycles.