2-distance (Δ+1)-coloring of sparse graphs using the potential method

03/22/2021

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by   Hoang La, et al.

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A 2-distance k-coloring of a graph is a proper k-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance (Δ+1)-coloring for graphs with maximum average degree less than 18/7 and maximum degree Δ≥ 7. As a corollary, every planar graph with girth at least 9 and Δ≥ 7 admits a 2-distance (Δ+1)-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on 2-distance coloring.