Locating chromatic number of powers of paths and cycles

Abstract

Let c be a proper k-coloring of a graph G. Let π = (R1, R2, .., Rk) be the partition of V(G) induced by c, where Ri is the partition class receiving color i. The color code cπ(v) of a vertex v of G is the ordered k-tuple (d(v, R1), d(v, R2), .., d(v, Rk)), where d(v, Ri) is the minimum distance from v to each other vertex u ∈ Ri for 1 ≤ i ≤ k. If all vertices of G have distinct color codes, then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by ΧL(G), is the smallest k such that G admits a locating coloring with k colors. In this paper, we give a characterization of the locating chromatic number of powers of paths. In addition, we find sharp upper and lower bounds for the locating chromatic number of powers of cycles.