Cores of Cubelike Graphs

TL;DR

This paper investigates the structure of cores of cubelike graphs, revealing they inherit many properties from the original graphs and are closely related to cubelike graphs, but it remains unresolved whether cores are always cubelike.

Contribution

It provides new structural, spectral, and group-theoretical insights into the cores of cubelike graphs, constraining their possible forms and eliminating small non-cubelike cores.

Findings

Cores inherit structural properties from cubelike graphs.

Orbital graphs of the core also resemble cubelike graphs.

Non-cubelike cores are excluded for graphs up to 32 vertices.

Abstract

A graph is $\textit{cubelike}$ if it is a Cayley graph for some elementary abelian $2$-group $\mathbb{Z}_2^n$. The core of a graph is its smallest subgraph to which it admits a homomorphism. More than ten years ago, Ne\v{s}et\v{r}il and \v{S}\'amal (On tension-continuous mappings. $\textit{European J. Combin.,}$ 29(4):1025--1054, 2008) asked whether the core of a cubelike graph is cubelike, but since then very little progress has been made towards resolving the question. Here we investigate the structure of the core of a cubelike graph, deducing a variety of structural, spectral and group-theoretical properties that the core "inherits" from the host cubelike graph. These properties constrain the structure of the core quite severely --- even if the core of a cubelike graph is not actually cubelike, it must bear a very close resemblance to a cubelike graph. Moreover we prove the much stronger result that not only are these properties inherited by the core of a cubelike graph, but also by the orbital graphs of the core. Even though the core and its orbital graphs look very much like cubelike graphs, we are unable to show that this is sufficient to characterise cubelike graphs. However, our results are strong enough to eliminate all non-cubelike vertex-transitive graphs on up to $32$ vertices as potential cores of cubelike graphs (of any size). Thus, if one exists at all, a cubelike graph with a non-cubelike core has at least $128$ vertices and its core has at least $64$ vertices.