The Locating-chromatic Number of Disjoint Union of Fan Graphs

Abstract

Let G = (V,E) is a connected graph and c is a k-coloring of G. The color class of G is the set of colored vertexs i, denoted by Ci for 1 <= i <= k. Let phi is a ordered partition from V (G) to independent color classes that is C1;C2; ...;Ck, with vertexs of Ci given color by i, 1 <= i <= k. Distance of a vertex v in V to Ci denoted by d(v,Ci) is min {d(v, x)|x in Ci}. The color codes of a vertex v in V is the ordered k-vector c(Phi|v) = (d(v,C1), d(v,C2), ..., d(v,Ck)) where d(v,Ci) = min {d(v, x | x in Ci)} for 1 <= i <= k. If distinct vertices have distinct color codes, then c is called a locating-coloring of G. The locating-chromatic number
XL(G) is the minimum number of colors in a locating-coloring of G. Let H is a disconnected graph and c is a k-coloring of H then induced partition of Phi from V(H). The coloring c is locating k-coloring of H if all vertices of H have distinct color codes. The locating-chromatic number of H, denoted by XL'(H), is the smallest k such that H admits a locating-coloring with k colors. In this paper, we study the locating-chromatic number of disjoint union of fan graphs.