Properly even harmonious labelings of disconnected graphs

A graph GG with qq edges is said to be harmonious if there is an injection ff from the vertices of GG to the group of integers modulo qq such that when each edge xyxy is assigned the label f(x)+f(y)(modq), the resulting edge labels are distinct. If GG is a tree, exactly one label may be used on two vertices. Over the years, many variations of harmonious labelings have been introduced.We study a variant of harmonious labeling. A function ff is said to be a properly even harmonious labeling of a graph GG with qq edges if ff is an injection from the vertices of GG to the integers from 0 to 2(q−1)2(q−1) and the induced function f∗f∗ from the edges of GG to 0,2,…,2(q−1)0,2,…,2(q−1) defined by f∗(xy)=f(x)+f(y)(mod2q) is bijective. This paper focuses on the existence of properly even harmonious labelings of the disjoint union of cycles and stars, unions of cycles with paths, unions of squares of paths, and unions of paths.