Product irregularity strength of graphs with small clique cover number

Abstract

For a graph X without isolated vertices and without isolated edges, a product-irregular labelling ω : E(X) → {1, 2, …, s}, first defined by Anholcer in 2009, is a labelling 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 exists a product irregular labeling is called the product irregularity strength of X and is denoted by ps(X). Clique cover number of a graph is the minimum number of cliques that partition its vertex-set. In this paper we prove that connected graphs with clique cover number 2 or 3 have the product-irregularity strength equal to 3, with some small exceptions.