On the well-coveredness of Cartesian products of graphs

A graph GG is well-covered   if every maximal independent set has the same cardinality. This paper investigates when the Cartesian product of two graphs is well-covered. We prove that if GG and HH both belong to a large class of graphs that includes all non-well-covered triangle-free graphs and most well-covered triangle-free graphs, then G×HG×H is not well-covered. We also show that if GG is not well-covered, then neither is G×GG×G. Finally, we show that G×GG×G is not well-covered for all graphs of girth at least 5 by introducing super well-covered graphs and classifying all such graphs of girth at least 5.