Harmonious order of graphs

We consider the following generalization of the concept of harmonious graphs. Given a graph G=(V,E)G=(V,E) and a positive integer t≥|E|t≥|E|, a function h̃:V(G)→Zt is called a tt-harmonious labeling   of GG if h̃ is injective for t≥|V|t≥|V| or surjective for t<|V|t<|V|, and h̃(v)+h̃(w)≠h̃(x)+h̃(y) for all distinct edges vw,xy∈E(G)vw,xy∈E(G). Then the smallest possible tt such that GG has a tt-harmonious labeling is named the harmonious order   of GG. We determine the harmonious order of some non-harmonious graphs, such as complete graphs KnKn (n≥5n≥5), complete bipartite graphs Km,nKm,n (m,n>1m,n>1), even cycles CnCn, some powers of paths Pnk, disjoint unions of triangles nK3nK3 (nn even). We also present some general results concerning harmonious order of the Cartesian product of two given graphs or harmonious order of the disjoint union of copies of a given graph. Furthermore, we establish an upper bound for harmonious order of trees.