Facial unique-maximum edge and total coloring of plane graphs

Abstract

An edge coloring of a plane graph is facial if every two facially adjacent edges (i.e., the edges that are adjacent and consecutive in a cyclic order around their common end vertex) receive different colors. A total coloring of a plane graph is facial if every two adjacent vertices, every two facially adjacent edges and every two incident elements receive different colors. A coloring of a plane graph (using linearly ordered color set) is unique-maximum if, for each face, the maximum color on its elements is used exactly once. In the paper it is proven, that every 2-edge-connected plane graph is facially unique-maximum 4-edge-colorable and facially unique-maximum 6-total-colorable, and that the bounds 4 and 6, respectively, are best possible. Furthermore, every plane graph is facially unique-maximum 6-edge-choosable and facially unique-maximum 8-total-choosable.