Difference of facial achromatic numbers between two triangular embeddings of a graph

Abstract

A facial $3$-complete $k$-coloring of a triangulation $G$ on a surface is a vertex $k$-coloring such that every triple of $k$-colors appears on the boundary of some face of $G$. The facial $3$-achromatic number $\psi_3(G)$ of $G$ is the maximum integer $k$ such that $G$ has a facial $3$-complete $k$-coloring. This notion is an expansion of the complete coloring, that is, a proper vertex coloring of a graph such that every pair of colors appears on the ends of some edge. For two triangulations $G$ and $G'$ on a surface, $\psi_3(G)$ may not be equal to $\psi_3(G')$ even if $G$ is isomorphic to $G'$ as graphs. Hence, it would be interesting to see how large the difference between $\psi_3(G)$ and $\psi_3(G')$ can be. We shall show that the upper bound for such difference in terms of the genus of the surface.