Odd facial total-coloring of maximal plane and outerplane graphs

A facial total-coloring of a plane graph G is a coloring of the vertices and edges such that no facially adjacent edges, 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 article, it is shown that every outerplane graph with triangular internal faces admits an odd facial total-coloring with at most 12 colors.In the case of maximal outerplane graphs, 9 colors are sufficient for such a coloring.It is also shown that every maximal plane graph has an odd facial total-coloring with 6 or 7 colors.All of the obtained bounds are tight.