Harmonious Labelings of Disconnected Graphs Involving Cycles and Multiple Components Consisting of Starlike Trees

Abstract

A harmonious labeling of a (simple) graph $G = (V,E)$ on $m>0$ edges is a one-to-one function $f:V \rightarrow \textbf{Z}_m$ such that if $e_1, e_2 \in E$ with respective endpoints $u_1,v_1$ and $u_2,v_2$, then $f(u_1) + f(v_1) \not \equiv f(u_2) + f(v_2) (\hbox {mod } m )$. If such a function exists, then G is said to be harmonious. If G were a tree, then precisely one vertex label is allowed to be used twice. A starlike tree is a tree with a central vertex adjacent to one endpoint of some number of paths each with the same number of vertices. It has been shown using cyclic groups that the disjoint union of an odd cycle on s vertices and starlike trees with the central vertex adjacent to some even $t\ge 2$ many s-paths is harmonious. We now consider the disjoint union of an odd cycle with at least two starlike trees with new notions of harmonious labelings to accommodate the case where $|V| >|E|$, one of which is a basic generalization of harmonious labeling and the other of which is a stricter and more balanced harmonious labeling.