Harmonious labeling of the union of m-cycle and n-path graph

A graph G is called harmonious graph if there is an injective mapping f from the set of vertices of G to the set Zq so that the mapping induced a bijection mapping f* from the set of edges of G to Zq, which is defined as f *(xy) = f (x) + f (y) for each edge xy∈ E(G). The mapping f is called harmonious labeling. Harmonious labeling concept has been developed since 1980. At that time, harmonious labeling is inspired by channel assignment problems. Nowadays, harmonious labeling has been known for various classes of graphs. However, there are still many certain classes of graphs that have not been determined yet whether they can be harmonious labeled or not. One of the class of graphs that have not been determined yet whether it can be harminous labeled or not is graph C_m ∪ P_n. In this paper, we prove that there is a harmonious labeling of graph C_m ∪ P_n for some values m and.