On well-covered Cartesian products

Bert L. Hartnell; Douglas F. Rall; Kirsti Wash

paper

On well-covered Cartesian products

arXiv:1703.08716

Abstract

In 1970, Plummer defined a well-covered graph to be a graph $G$ in which all maximal independent sets are in fact maximum. Later Hartnell and Rall showed that if the Cartesian product $G □ H$ is well-covered, then at least one of $G$ or $H$ is well-covered. In this paper, we consider the problem of classifying all well-covered Cartesian products. In particular, we show that if the Cartesian product of two nontrivial, connected graphs of girth at least $4$ is well-covered, then at least one of the graphs is $K_{2}$ . Moreover, we show that $K_{2} □ K_{2}$ and $C_{5} □ K_{2}$ are the only well-covered Cartesian products of nontrivial, connected graphs of girth at least $5$ .

12 pages, 2 figures