Odd Facial Total-Coloring of Unicyclic Plane Graphs

A facial total-coloring of a plane graph G is a coloring of the vertices and edges such that no facially adjacent edges (edges that are consecutive on the boundary walk of a face of G ), no adjacent vertices, and no edge and its endvertices are assigned the same color. A facial total-coloring of G is odd if for every face f and every color c , either no element or an odd number of elements incident with f is colored by c . In this paper, it is proved that every unicyclic plane graph admits an odd facial total-coloring with at most 10 colors. It is also shown that this bound is tight.