Facial Achromatic Number of Triangulations with Given Guarding Number

Authors

Naoki Matsumoto, Keio UniversityFollow
Yumiko OHNO, Yokohama National UniversityFollow

Publication Date

February 2022

Abstract

A (not necessarily proper) k-coloring c : V(G) → {1,2,…k} of a graph G on a surface is a facial t-complete k-coloring if every t-tuple of colors appears on the boundary of some face of G. The maximum number k such that G has a facial t-complete k-coloring is called a facial t-achromatic number of G, denoted by ψ_t(G). In this paper, we investigate the relation between the facial 3-achromatic number and guarding number of triangulations on a surface, where a guarding number of a graph G embedded on a surface, denoted by guard(G), is the smallest size of its guarding set which is a generalized concept of guards in the art gallery problem. We show that for any graph G embedded on a surface, ψ Δ(G^*)(G) ≤ guard(G) + Δ(G*) – 1, where Δ(G*) is the largest face size of G. Furthermore, we investigate sufficient conditions for a triangulation G on a surface to satisfy ψ₃(G) = guard(G) + 2. In particular, we prove that every triangulation G on the sphere with guard(G) = 2 satisfies the above equality and that for one with guarding number 3, it also satisfies the above equality with sufficiently large number of vertices.

Creative Commons License

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

Recommended Citation

Matsumoto, Naoki and OHNO, Yumiko (2022) "Facial Achromatic Number of Triangulations with Given Guarding Number," Theory and Applications of Graphs: Vol. 9: Iss. 1, Article 1.
DOI: 10.20429/tag.2022.090101
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol9/iss1/1