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

Source abstract

<p>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 $\chi _i^l(G) \le \Delta + 4$ if $\Delta \ge 12$ 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 $\chi _i^l(G) \le \Delta + 4$ when $\Delta \ge 12$ and $G$ has disjoint ${5^ - }$-cycles.</p>