Odd Facial Total-Coloring of Unicyclic Plane Graphs
Július Czap · Discrete Mathematics Letters · 2022
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.