Injective (Δ + 1)-coloring of planar graphs with girth 6

A vertex coloring of a graph G is called injective if every two vertices joined by a path of length 2 get different colors. The minimum number χi(G) of the colors required for an injective coloring of a graph G is clearly not less than the maximum degree Δ(G) of G. There exist planar graphs with girth g ≥ 6 and χi = Δ+1 for any Δ ≥ 2. We prove that every planar graph with Δ ≥ 18 and g ≥ 6 has χi ≤ Δ + 1.

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