Further Results on Locating Chromatic Number for Amalgamation of Stars Linking by One Path

Abstract

Let G = (V,E) be a connected graph. Let c be a proper coloring using k colors, namely 1, 2,·s, k. Let P={S₁, S₂,..., S_k} be a partition of V(G) induced by c and let S_i be the color class that receives the color i. The color code, c_P(v)=(d(v,S₁), d(v,S₂),...,d(v,S_k)), where d(v,S_i)=min {d(v,x)|x Î S_i} for i Î [1,k]. If all vertices in V(G) have different color codes, then c is called as the \emphlocating-chromatic k-coloring of G. Minimum k such that G has the locating-chromatic k-coloring is called the locating-chromatic number, denoted by c_L(G). In this paper, we discuss the locating-chromatic number for n certain amalgamation of stars linking a path, denoted by nS_k,m, for n ≥ 1, m ≥ 2, k ≥ 3, and k>m.