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

TL;DR

This paper proves that sparse graphs with certain average degree constraints and sufficiently large maximum degree can be properly 2-distance colored with +2 colors, extending to planar graphs with large girth.

Contribution

It establishes the existence of +2 coloring for graphs with specific average degree and degree bounds, including planar graphs with large girth.

Findings

Graphs with max average degree < 8/3 and   +2 coloring.

Graphs with max average degree < 14/5 and   +2 coloring.

Planar graphs with girth  or  and large degree admit +2 coloring.

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 $\Delta\geq 6$ (resp. $\Delta\geq 10$). As a corollary, every planar graph with girth at least $8$ (resp. $7$) and maximum degree $\Delta\geq 6$ (resp. $\Delta\geq 10$) admits a $2$-distance $(\Delta+2)$-coloring.