Improved 2-distance coloring of planar graphs with maximum degree 5

Abstract

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two or less get different colors. The 2-distance chromatic number of G is the minimum k such that G has a 2-distance k-coloring, denoted by ${\chi }_2\left(G\right)$. In this paper, we show that ${\chi }_2\left(G\right)\le 17$ for every planar graph G with maximum degree $\Delta \left(G\right)\le 5$, which improves a former bound ${\chi }_2\left(G\right)\le 18$.