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, ⋯, k. Let Π = {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_Π(v) = (d(v, S₁), d(v, S₂), ⋯, d(v, S_k)), where d(v, S_i) = min{d(v, x)∣x ∈ S_i} for i ∈ [1, k]. If all vertices in V(G) have different color codes, then c is called as the locating-chromatic k-coloring of G. Minimum k such that G has the locating-chromatic k-coloring is called the locating-chromatic number, denoted by χ_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.