Cayley graphs on elementary abelian groups of extreme degree have complete cores
Guang Rao, Colin Tan
TL;DR
This work establishes a sharp extremal criterion for when a Cayley graph on an elementary abelian $p$-group has a complete core induced by an $\mathbb{F}_p$-subspace. By introducing the Complete Core Axiom (CCA), the authors derive a general mechanism to obtain complete cores from direct-sum decompositions and projection retractions, and they prove a central theorem: for $X=\mathrm{Cay}(({\mathbb F}_p)^d, C)$, if $\deg(X)<\kappa(p)$ or $\deg(X)\ge |V(X)|-\kappa(p)$, then $X^\bullet$ is complete and equals $X|_V=\mathrm{Cay}(V, V\setminus\{0\})$ for some $\mathbb{F}_p$-subspace $V$ of dimension at least $d-2$ when the high-degree case occurs. The bounds are demonstrated to be sharp via explicit counterexamples for all primes, including the folded $5$-cube for $p=2$ and a projective-space construction for $p=3$, while Hahn–Tardif-type results ensure cores for $p\ge5$. The paper also extends these results to $p$-ary Cayley graphs for odd primes and discusses open questions about intermediate-degree graphs. The methods combine structural decompositions, projections, and combinatorial case analysis to classify core behavior in extreme-degree regimes.
Abstract
Nešetřil and Šámal asked whether every cubelike graph has a cubelike core. Mančinska, 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 Nešetřil and Šámal'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 $p$.