Injective coloring of planar graphs with girth 5

A $k$-injective-coloring of a graph $G$ is a mapping $c:V\left(G\right)\to \left\{1,2,\cdots ,k\right\}$ such that $c\left(u\right)\ne c\left(v\right)$ for any two vertices $u$ and $v$ if $u$ and $v$ have a common vertex. The injective chromatic number of $G$, denoted by ${\chi }_i\left(G\right)$, is the least $k$ such that $G$ has an injective k-coloring. In this paper, we prove that for planar graph $G$ with $g\left(G\right)\ge 5$, $\mathrm{\Delta }\left(G\right)\ge 20$ and without adjacent 5-cycles, ${\chi }_i\left(G\right)\le \mathrm{\Delta }\left(G\right)+2$.