Locating-chromatic number for a graph of two components

Abstract

The study of locating-chromatic number of a graph initiated by Chartrand et al. [5] is only limited for connected graphs. In 2014, Welyyanti et al. extended this notion so that the locating-chromatic number can also be applied to disconnected graphs. Let c be a k-coloring of a disconnected graph H(V, E) and ∏ = {C1,C2, …, Ck} be the partition of V (H) induced by c, where Ci is the set of all vertices receiving color i. The color code c∏(v) of a vertex v ∈ H is the ordered k-tuple (d(v,C1), d(v,C2), …, d(v,Ck)), where d(v,Ci) = min{d(v, x)|x ∈ Ci} and d(v,Ci) < ∞ for all i ∈ [1, k]. If all vertices of H have distinct color codes, then c is called a locating-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, otherwise we say that χ′L(H)=∞. In this paper, we determine locating-chromatic number of a graph with two components where each component has the locating-chromatic number 3.