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

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 $p$.