Physical Sciences and Mathematics Commons^™

Difference Of Facial Achromatic Numbers Between Two Triangular Embeddings Of A Graph, Kengo Enami, Yumiko Ohno

Theory and Applications of Graphs

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 $G$ …

Facial Achromatic Number Of Triangulations With Given Guarding Number, Naoki Matsumoto, Yumiko Ohno

Theory and Applications of Graphs

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, …