Difference of Facial Achromatic Numbers between Two Triangular Embeddings of a Graph

Authors

Kengo Enami, Seikei UniversityFollow
Yumiko Ohno, Yokohama National UniversityFollow

Publication Date

December 2023

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 an upper bound for such difference in terms of the genus of the surface.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Enami, Kengo and Ohno, Yumiko (2023) "Difference of Facial Achromatic Numbers between Two Triangular Embeddings of a Graph," Theory and Applications of Graphs: Vol. 10: Iss. 2, Article 6.
DOI: 10.20429/tag.2023.10206
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol10/iss2/6