Even harmonious labelings of disjoint graphs with a small component

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. We investigate the existence of properly even harmonious labelings of families of disconnected graphs with one of C ₃ , C ₄ , K ₄ or W ₄ as a component.