Cores of Cayley graphs

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A homomorphism $f:G\to H$ of finite simple graphs is a vertex map satisfying $f(u)f(v)\in E(H)$ whenever $uv\in E(G)$. A core of $G$ is a graph of minimum vertex cardinality among graphs admitting homomorphisms both from and to $G$; it is unique up to isomorphism. For a finite abelian group $M$, its power $M^a$ is the direct product of $a$ copies. For an inverse-closed set $S\subseteq M^a\setminus\{0\}$, the Cayley graph $\operatorname{Cay}(M^a,S)$ has vertices $M^a$, with $x,y$ adjacent when $y-x\in S$. The exponent $a=0$ is allowed, with a one-element group. Cayley graphs on powers of $\mathbb Z/2\mathbb Z$ are called cubelike graphs.

Statement recorded in the supplied source.

The source asks whether the core of every Cayley graph on a power of $M$ is isomorphic to a Cayley graph on some power of the same group $M$. It singles out $M=\mathbb Z/2\mathbb Z$. Its discussion additionally proposes restricting $M$ to $\mathbb Z/p\mathbb Z$ for primes $p$, or replacing the conclusion by a Cayley graph on a homomorphic image of the original group. [S2]

Short English statement

Are cores of Cayley graphs again Cayley graphs on powers of the same abelian group, especially elementary abelian groups, or on group quotients?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3206 (OPG-125). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Cores of Cayley graphs, original node 125, statement and mathematical discussion. https://www.openproblemgarden.org/op/cores_of_cayley_graphs.