Harmonious Labelings of Families of Disjoint Unions of an Odd Cycle and Certain Trees

A graph \ (G\) with vertex set \ (V = V (G) \) and edge set \ (E = E (G) \) is harmonious if there exists a harmonious labeling of \ (G\) ; which is an injective function \ (f: V (G) Zₘ\) provided that whenever \ (e₁, e₂ E\) are distinct with endpoints \ (u₁, v₁\) and \ (u₂, v₂\), respectively, then \ (f (u₁) + f (v₁) f (u₂) + f (v₂) (mod m) \). Using basic group theory, we prove in a different manner an already established result that a disjoint union of an odd cycle and a path is harmonious provided their lengths satisfy certain conditions. We apply the same basic idea to establish that, under the same conditions, a disjoint union of an odd cycle with a certain starlike tree is harmonious (where a starlike tree consists of a central vertex that is adjacent to an endpoint of a certain number of fixed length paths). Finally, we extend the latter result to include specifying that the central vertex in the tree be adjacent to different vertices in each of the \ (t\) -many \ (s\) -paths.