On locating-chromatic number of helm graph H_m FOR 10≤m≤28

Abstract

Let G = (V,E) be a connected graph and c be a k-coloring of G. The color class S_i of G is a set of vertices given color i, for 1 ≤ i ≤ k. Let  = {S_1,S_2,…,S_k} be an ordered partition of V(G). The color code of a vertex v $in$ (element) V(G) is defined as the ordered k--tuple

c_Π (v)=(d(v,S_1),d(v,S_2),...,d(v,S_k)),

where d(v,S_i) = min{d(v,x)| x $in$ (element) S_i} for 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then c is called a locating-coloring of G. The locating-chromatic number χ_L (G) is the minimum number of colors in a locating-coloring of G. This paper discusses the locating-chromatic number of helm graph H_m for 10 ≤ m ≤ 28. Helm graph H_m is constructed by adding some leaves to the corresponding vertices of wheels W_m, for m ≥ 3.