2-Distance Coloring of 4-Irregular Planar Graphs

Abstract

A $k$-irregular graph is a graph with maximum degree $k$ such that vertices of degree $k$ are not adjacent. A $2$-distance $k$-coloring of a graph is a coloring of the vertices using $k$ colors in which any two vertices at distance at most $2$ receive distinct colors. The $2$-distance chromatic number of $G$, denoted by $χ_{2}(G)$, is the minimum integer $k$ such that $G$ admits a $2$-distance $k$-coloring. Zhu \cite{zhu} proved that $χ_2(G)\leq 13$ for planar graphs with maximum degree at most $4$. We prove that for a 4-irregular planar graph $G$, we have $χ_2(G) \leq 10$.