Facial Achromatic Number of Triangulations with Given Guarding Number

Abstract

A (not necessarily proper) k -coloring c : V ( G ) ! f 1 ; 2 ; : : : ; k g 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 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, where ∆( G (cid:3) ) is the largest face size of G . Furthermore, we investigate sufficient conditions for a triangulation G on a surface to satisfy 3 ( G ) = guard( G ) + 2. In particular, we prove that every triangulation G on the sphere with guard( G ) = 2 satis(cid:12)es the above equality and that for one with guarding number 3, it also satis(cid:12)es the above equality with sufficiently large number of vertices.