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

Abstract

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 187 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.