The Locating Chromatic Number of the Cyclic Chain Graph

Abstract

The locating chromatic number of graph G (_χL(G)) combines the idea of the partition dimension and the chromatic number by considering the locations of the vertices of graph G. Let (C_ni, m) be a cyclic chain graph, namely a group of blocks in the form of a cycle graph C_n1⁽¹⁾, C_n2⁽²⁾, ···, C_ni⁽ⁱ⁾. The ni is the number of vertices on the i-th cycle, and m is the number of cycles, for _ni ≥ 3, 1 ≤ i ≤ m, and m ≥ 2, and the vertex v_i,_⌈ni/2⌉+1 in C_ni⁽ⁱ⁾ is identified with the vertex v_i,_⌈ni/2⌉+1 in C_ni+1⁽ⁱ⁺¹⁾. In this research, we determine _χL(C_ni, m) for _ni ≥ 3, 1 ≤ i ≤ m, and m ≥ 2.