The locating chromatic number for m-shadow of a connected graph

Abstract

Let c : V ( G )→{1, 2, …, k } be a proper k -coloring of a simple connected graph G . Let Π = { C 1 , C 2 , …, C k } be a partition of V ( G ) , where C i is the set of vertices of G receiving color i . The color code, c Π ( v ) , of a vertex v with respect to Π is an ordered k -tuple ( d ( v , C 2 ), d ( v , C 2 ),…, d ( v , C k )) , where d ( v , C i )=min{ d ( v , x ): x ∈ C i } for i = 1, 2, …, k . If distinct vertices have distinct color codes then c is called a locating coloring of G . The minimum k for which c is a locating coloring is the locating chromatic number of G , denoted by χ L ( G ) . Let G be a non trivial connected graph and let m ≥ 2 be an integer. The m - shadow of G , denoted by D m ( G ) , is a graph obtained by taking m copies of G , say G 1 , G 2 , …, G m , and each vertex v in G i , i = 1, 2, …, m − 1 , is joined to the neighbors of its corresponding vertex v ′ in G i + 1 . In the present paper, we deal with the locating chromatic number for m -shadow of connected graphs. Sharp bounds on the locating chromatic number of D m ( G ) for any non trivial connected graph G and any integer m ≥ 2 are obtained. Then the values of locating chromatic number for m -shadow of complete multipartite graphs and paths are determined, some of which are considered to be optimal.