On Bounds for the Product Irregularity Strength of Graphs

Abstract

For a graph $$X$$X with at most one isolated vertex and without isolated edges, a product-irregular labeling$$\omega :E(X)\rightarrow \{1,2,\ldots ,s\}$$?:E(X)?{1,2,?,s}, first defined by Anholcer in 2009, is a labeling of the edges of $$X$$X such that for any two distinct vertices $$u$$u and $$v$$v of $$X$$X the product of labels of the edges incident with $$u$$u is different from the product of labels of the edges incident with $$v$$v. The minimal $$s$$s for which there exist a product irregular labeling is called the product irregularity strength of $$X$$X and is denoted by $$ps(X)$$ps(X). In this paper it is proved that $$ps(X)\le |V(X)|-1$$ps(X)≤|V(X)|-1 for any graph $$X$$X with more than $$3$$3 vertices. Moreover, the connection between the product irregularity strength and the multidimensional multiplication table problem is given, which is especially expressed in the case of the complete multipartite graphs.