A note on 2-facial coloring of plane graphs

An l-facial coloring of a plane graph is a vertex coloring such that any two different vertices joined by a facial walk of length at most l receive distinct colors. It is known that every plane graph admits a 2-facial coloring using 8 colors [D. Král, T. Madaras, R. Škrekovski, Cyclic, diagonal and facial coloring, European J. Combin. 3–4 (26) (2005) 473–490]. We improve this bound for plane graphs with large girth and prove that if G is a plane graph with girth g⩾14 (resp. 10, 8) then G admits a 2-facial coloring using 5 colors (resp. 6, 7). Moreover, we give exact bounds for outerplanar graphs and K4-minor free graphs.