Even harmonious labelings of disjoint graphs with a small component

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. We investigate the existence of properly even harmonious labelings of families of disconnected graphs with one of C3,C4,K4C3,C4,K4 or W4W4 as a component.