List injective coloring of planar graphs with girth 5, 6, 8

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