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

TL;DR

This paper investigates the maximum difference in facial 3-achromatic numbers between two different triangular embeddings of the same graph on a surface, establishing an upper bound related to the surface's genus.

Contribution

It introduces the concept of facial 3-achromatic number for triangulations and derives an upper bound for its variation across different embeddings of the same graph.

Findings

Maximum difference in facial 3-achromatic numbers depends on the surface's genus.

The paper provides an explicit upper bound for this difference.

It extends the understanding of coloring properties in different embeddings of the same 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.