$2$-distance $(\Delta +2)$-coloring of sparse graphs

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 ($\Delta +2$)-coloring for graphs with maximum average degree less than $\frac{8}{3}$ (resp. $\frac{14}{5}$) and maximum degree $\Delta \ge 6$ (resp. $\Delta \ge 10$). As a corollary, every planar graph with girth at least $8$ (resp. $7$) and maximum degree $\Delta \ge 6$ (resp. $\Delta \ge 10$) admits a $2$-distance $(\Delta +2)$-coloring.

14 pages, 20 figures. arXiv admin note: substantial text overlap with arXiv:2103.11687, arXiv:2106.03587, arXiv:2105.01684