List injective colorings of planar graphs

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 @f such that @f(v)@?L(v) whenever [email protected]?V(G). The least k for which G is injectively k-choosable is denoted by @g"i^l(G). Note that @g"i^l>[email protected] for every graph with maximum degree @D. For planar graphs with girth g, Bu et al. (2009) [15] proved that @g"i^[email protected] if @D>=71 and g>=7, which we strengthen here to @D>=16. On the other hand, there exist planar graphs with g=6 and @g"i^[email protected]+1 for any @D>=2. Cranston et al. (submitted for publication) [16] proved that @g"i^[email protected][email protected]+1 if g>=9 and @D>=4. We prove that each planar graph with g>=6 and @D>=24 has @g"i^[email protected][email protected]+1.

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