Properly even harmonious labelings of disconnected graphs

A graph G with q edges is said to be harmonious if there is an injection f from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x). +. f(y). (mod q), the resulting edge labels are distinct. If G 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 f is said to be a properly even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2(q-1) and the induced function f ^* from the edges of G to 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.