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) (mod 2q) is bijective. We investigate the existence of properly even harmonious labelings of families of disconnected graphs with one of C3, C4, K4 or W4 as a component.