Complete bipartite graph is a totally irregular total graph

Abstract

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 K m , n , the t s ( K 1, n ) and the t s ( K n , n ) are already determined for any positive integer n. Completing the results, this paper deals with the total irregularity strength of complete bipartite graph K m , n for any positive integer m and n.