On Bounds for the Product Irregularity Strength of Graphs

For a graph with at most one isolated vertex and without isolated edges, a product-irregular labeling , first defined by Anholcer in 2009, is a labeling of the edges of such that for any two distinct vertices and of the product of labels of the edges incident with is different from the product of labels of the edges incident with . The minimal for which there exist a product irregular labeling is called the product irregularity strength of and is denoted by . In this paper it is proved that for any graph with more than 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.