Locating-chromatic number of amalgamation of stars

Abstract

Let G be a connected graph and c a proper coloring of G. For i=1,2,...,k define the color class Ci as the set of vertices receiving color i. The color code cΠ(v) of a vertex v in G is the ordered k-tuple (d(v,C1),...,d(v,Ck)) where d(v,C1) is the distance of v to Ci. If all distinct vertices of G have distinct color codes, then c is called a locating-coloring of G. The locating-chromatic number of graph G, denoted by χL(G) is the smallest k such that G has a locating coloring with k colors. In this paper we discuss the locating-chromatic number of amalgamation of stars Sk,m. Sk,m is obtained from k copies of star K1,m by identifying a leaf from each star. We also determine a sufficient condition for a connected subgraph H⊆ Sk,m satisfying χL(H) ≤ χL(Sk,m).