FACIAL TOTAL-COLORING OF BIPARTITE PLANE GRAPHS
Július Czap, Peter Šugerek · International Journal of Pure and Apllied Mathematics · 2017
Let G be a plane graph.Two edges are facially adjacent in G if they are consecutive edges on a boundary walk of a face of G.A facial edge-coloring of G is an edge-coloring such that any two facially adjacent edges receive distinct colors.A facial totalcoloring of G is a coloring of vertices and edges such that no facially adjacent edges, no adjacent vertices, and no edge and its endvertices are assigned the same color.In this paper we prove that every plane graph admits a facial edge-coloring with at most four colors such that at most three colors appear at each vertex.Using this result we confirm a conjecture on facial total-coloring for bipartite plane graphs.