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

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