3-Facial Coloring of Plane Graphs

Abstract

A plane graph is l-facially $k$-colorable if its vertices can be colored with $k$ colors such that any two distinct vertices on a facial segment of length at most lare colored differently. We prove that every plane graph is $3$-facially $11$-colorable. As a consequence, we derive that every $2$-connected plane graph with maximum face-size at most $7$ is cyclically $11$-colorable. These two bounds are just one higher than those that are proposed by the $(3\l+1)$-conjecture and the cyclic conjecture.