Well-Covered Powers: Cartesian Products and Cayley Graphs

arXiv:2611.03956v1 Announce Type: new Abstract: We study graphs whose powers are well-covered, with an emphasis on Cartesian products and Cayley graphs. We first establish necessary conditions and sufficient criteria for Cartesian products, provide examples showing that a factor need not be well-covered and that two noncomplete factors can have the well-covered powers property, and classify several structured product families. In particular, we construct graphs satisfying the well-covered powers property for arbitrarily large diameters. We then show that the Cartesian square of a connected abelian Cayley graph is well-covered exactly when the graph is complete. We classify the well-covered-power graphs among connected circulant graphs of degree at most four. For Cartesian products of any number of complete graphs, we also prove that the recursive alphabet-size inequalities in the final conjecture are sufficient for all powers to be well-covered, prove partial converses for arbitrarily many factors, and give explicit obstructions for every binary prefix followed by one complete factor.

You are being forwarded in 6s — stay on this page