On Bounds for the Product Irregularity Strength of Graphs

Abstract

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