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List injective colorings of planar graphs. (English) Zbl 1216.05027
Summary: A vertex coloring of a graph $$G$$ is called injective if any two vertices joined by a path of length two get different colors. A graph $$G$$ is injectively $$k$$-choosable if any list $$L$$ of admissible colors on $$V(G)$$ of size $$k$$ allows an injective coloring $$\varphi$$ such that $$\varphi(v)\in L(v)$$ whenever $$v\in V(G)$$. The least $$k$$ for which $$G$$ is injectively $$k$$-choosable is denoted by $$\chi^l_i(G)$$.
Note that $$\chi^l_i\geq \Delta$$ for every graph with maximum degree $$\Delta$$. For planar graphs with girth $$g$$, Y. Bu, D. Chen, A. Raspaud and W. Wang [Discrete Appl. Math. 157, No. 4, 663–672 (2009; Zbl 1173.05320)] proved that $$\chi^l_i= \Delta$$ if $$\Delta\geq 71$$ and $$g\geq> 7$$, which we strengthen here to $$\Delta\geq 16$$. On the other hand, there exist planar graphs with $$g= 6$$ and $$\chi^l_i= \Delta+ 1$$ for any $$\Delta\geq 2$$. D. W. Cranston, S.-J. Kim and G. Yu [Discrete Math. 310, No. 21, 2965–2973 (2010; Zbl 1209.05075)] proved that $$\chi^l_i\leq\Delta+ 1$$ if $$g\geq 9$$ and $$\Delta\geq 4$$. We prove that each planar graph with $$g\geq 6$$ and $$\Delta\geq 24$$ has $$\chi^l_i\leq\Delta+ 1$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
##### Keywords:
planar graph; injective coloring; girth
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##### References:
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