Abstract
A vertex coloring of a graph G is called injective if any two vertices with a common neighbor receive distinct colors. Let χi(G), χil(G) be the injective chromatic number and injective choosability number of G, respectively. Suppose that G is a planar graph with maximum degree Δ and girth g. We show that (1) if g≥5 then χil(G)≤Δ+7 for any Δ, and χil(G)≤Δ+4 if Δ≥13; (2) χil(G)≤Δ+2 if g≥6 and Δ≥8; (3) χil(G)≤Δ+1 if g≥8 and Δ≥5.
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