On the list injective coloring of planar graphs without a $4^{-}$-cycle intersecting with a $5^{-}$-cycle

An injective coloring of a graph $G$ is a vertex coloring such that a pair of vertices obtain distinct colors if there is a path of length two between them. It is proved in this paper that ${\chi }_i^l(G)\le \mathrm{\Delta }+4$ if $\mathrm{\Delta }\ge 12$ when $G$ does not have a $4^{-}$-cycle intersecting with a $5^{-}$-cycle. Our result improves a previous result of Cai et al. in 2023, who showed that ${\chi }_i^l(G)\le \mathrm{\Delta }+4$ when $\mathrm{\Delta }\ge 12$ and $G$ has disjoint $5^{-}$-cycles.