Even harmonious 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. When G is a tree, exactly one label may be used on two vertices. Recently two variants of harmonious labelings have been defined. A function f is said to be an odd 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 2q - 1 such that the induced mapping f*(uv) = f(u) + f(v) from the edges of G to the odd integers between 1 to 2q - 1 is a bijection. A function f is said to be an 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 2q and the induced function f* from the edges of G to {0, 2,...,2(q - 1)}g defined by f*(uv) = f(u) + f(v) (mod 2q) is bijective. In this paper we investigate the existence of even harmonious labelings for a number of common graph families. Special attention is given to disconnected graphs. We state some conjectures and open problems.