Harmonious order of graphs

We consider the following generalization of the concept of harmonious graphs. Given a graph G = ( V , E ) and a positive integer t ≥ | E | , a function h ̃ : V ( G ) → Z t is called a t -harmonious labeling of G if h ̃ is injective for t ≥ | V | or surjective for t < | V | , and h ̃ ( v ) + h ̃ ( w ) ≠ h ̃ ( x ) + h ̃ ( y ) for all distinct edges v w , x y ∈ E ( G ) . Then the smallest possible t such that G has a t -harmonious labeling is named the harmonious order of G . We determine the harmonious order of some non-harmonious graphs, such as complete graphs K n ( n ≥ 5 ), complete bipartite graphs K m , n ( m , n > 1 ), even cycles C n , some powers of paths P n k , disjoint unions of triangles n K 3 ( n 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.