Complete bipartite graph is a totally irregular total graph

A graph G is called a totally irregular total k -graph if it has a totally irregular total k-labeling λ : V ∪ E→ 1, 2, ... , k, that is a total labeling such that for any pair of different vertices x and y of G, their weights wt(x) and wt(y) are distinct, and for any pair of different edges e and f of G, their weights wt(e) and wt(f) are distinct. The minimum value k under labeling λ is called the total irregularity strength of G, denoted by ts(G). For special cases of a complete bipartite graph Km, n , the ts(K1, n) and the ts(Kn, n) are already determined for any positive integer n. Completing the results, this paper deals with the total irregularity strength of complete bipartite graph Km, n for any positive integer m and n.