Locating chromatic number of powers of paths and cycles (Q2335071)

Summary: Let \(c\) be a proper \(k\)-coloring of a graph \(G\). Let \(\pi = \{R_1, R_2, \ldots, R_k \}\) be the partition of \(V(G)\) induced by \(c\), where \(R_i\) is the partition class receiving color \(i\). The color code \(c_\pi(v)\) of a vertex \(v\) of \(G\) is the ordered \(k\)-tuple \((d(v, R_1), d(v, R_2), \ldots,\)\(d(v, R_k)),\) where \(d(v, R_i)\) is the minimum distance from \(v\) to each other vertex \(u \in R_i\) for \(1 \leq i \leq k\). 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 \(\chi_L(G)\), 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.