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

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<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>