Product irregularity strength of graphs with small clique cover number

TL;DR

This paper determines the product irregularity strength for connected graphs with small clique cover numbers, showing it is generally 3, with some exceptions, thus advancing understanding of graph labelings.

Contribution

It establishes that connected graphs with clique cover numbers 2 or 3 typically have a product irregularity strength of 3, clarifying this property for a class of graphs.

Findings

Connected graphs with clique cover number 2 or 3 have ps(X)=3.

Most such graphs have the minimal product irregularity strength of 3.

Some small exceptions are identified where the strength differs.

Abstract

For a graph $X$ without isolated vertices and without isolated edges, a product-irregular labelling $\omega:E(X)\rightarrow \{1,2,\ldots,s\}$, 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 exist 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.