Cayley graphs on elementary abelian groups of extreme degree have complete cores

TL;DR

This paper proves that certain cubelike and elementary abelian p-group Cayley graphs have complete cores under specific degree and size conditions, extending previous results and answering open questions.

Contribution

It extends the class of cubelike graphs known to have complete cores, including those with degree less than 5 or significantly smaller than the number of vertices, and generalizes results to elementary abelian p-group Cayley graphs.

Findings

Cores of cubelike graphs with degree less than 5 are complete.

Cores of cubelike graphs with degree at least 5 less than the number of vertices are complete.

Analogous results hold for Cayley graphs on elementary abelian p-groups for odd primes p.

Abstract

Ne\v{s}et\v{r}il and \v{S}\'{a}mal asked whether every cubelike graph has a cubelike core. Man\v{c}inska, Pivotto, Roberson and Royle answered this question in the affirmative for cubelike graphs whose core has at most $32$ vertices. When the core of a cubelike graph has at most $16$ vertices, they gave a list of these cores, from which it follows that every cubelike graph with degree strictly less than $5$ has a complete core. We prove the following extension: if the degree of a cubelike graph is either strictly less than $5$ or at least $5$ less than the number of its vertices, then its core is complete and induced by a $\mathbb{F}_2$-vector subspace of its vertices. Thus we also answer Ne\v{s}et\v{r}il and \v{S}\'{a}mal's question in the affirmative for cubelike graphs with degree at least $5$ less than the number of vertices. Our result is sharp as the $5$-regular folded $5$-cube and its graph complement are both non-complete cubelike graph cores. We also prove analogous results for Cayley graphs on elementary abelian $p$-groups for odd primes $p$.