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 \(G \Box H\) 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 \(K_2 \Box K_2\) and \(C_5 \Box K_2\) are the only well-covered Cartesian products of nontrivial, connected graphs of girth at least 5.
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The second author is supported by a grant from the Simons Foundation (Grant Number 209654 to Douglas F. Rall).
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Hartnell, B.L., Rall, D.F. & Wash, K. On Well-Covered Cartesian Products. Graphs and Combinatorics 34, 1259–1268 (2018). https://doi.org/10.1007/s00373-018-1943-3
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DOI: https://doi.org/10.1007/s00373-018-1943-3