The 2-Distance Chromatic Number of Planar Graphs Without 3,4,8-Cycles

For a graph G, a 2-distance k-coloring of G is a mapping \(\varphi :V(G)\rightarrow \{1,2,\ldots ,k\}\) $\phi :V(G)\to \{1,2,\dots ,k\}$ such that \(\varphi (u)\ne \varphi (v)\) $\phi (u)\ne \phi (v)$ for any two vertices u, v at distance at most two. The 2-distance chromatic number is the smallest integer k such that G has a 2-distance k-coloring, denoted by \(\chi _{2}(G)\) ${\chi }_2(G)$ . Wang and Lih (SIAM J Discrete Math 17:264–275, 2003) posed a famous conjecture which states that \(\chi _{2}(G)=\Delta (G)+1\) ${\chi }_2(G)=\mathrm{\Delta }(G)+1$ for a planar graph G with girth \(g\ge 5\) $g\ge 5$ and maximum degree \(\Delta (G)\ge M(g)\) $\mathrm{\Delta }(G)\ge M(g)$ . In this paper, we prove that every planar graph without 3,4,8-cycles and \(\Delta (G)\ge 18\) $\mathrm{\Delta }(G)\ge 18$ satisfies \(\chi _{2}(G)\le \Delta (G)+3\) ${\chi }_2(G)\le \mathrm{\Delta }(G)+3$ . This improves a result due to Bu and Yan (Adv Math (China) 44:208–218, 2015) who gave the upper bound \(\Delta (G)+5\) $\mathrm{\Delta }(G)+5$ when \(\Delta (G)\ge 14\) $\mathrm{\Delta }(G)\ge 14$ .