On Locating-chromatic Number for Graphs with Dominant Vertices

Abstract

Abstract Let c be a k -coloring of a (not necessary connected) graph H . Let П= { C 1, C 2, · · ·, Ck } be the partition of V ( H ) induced by c , where Ci is partition class receiving color i . The color code c П ( v ) of a vertex v ∈ H is the ordered k -tuple ( d ( v , C 1), d ( v , C 2), · · ·, d ( v , Ck )), where d ( v , Ci ) = min { d ( v , x )| x ∈ Ci } for all i ∈ [1, k ]. If all vertices of H have distinct color codes, then c is called a locating k-coloring of H . The locating-chromatic number of H , denoted by χ L ’ ( H ), is the smallest k such that H admits a locating- coloring with k colors. If there is no integer k satisfying the above conditions, then we say that χ’ L ( H ) = ∞. Note that if H is a connected graph, then χ L ’ ( H ) = χL ( H ). In this paper, we provide upper bounds for the locating-chromatic numbers of connectedgraphs obtained from disconnected graphs where each component contains a single dominant vertex.