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 (∆ + 1)-coloring for graphs with
maximum average degree less than 18 and maximum degree ∆ ≥ 7. As a corollary, every planar graph with 7
girth at least 9 and ∆ ≥ 7 admits a 2-distance (∆ + 1)-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on 2-distance coloring.