Intermediate-degree complete-core classification for p-ary Cayley graphs

Determine whether, for a fixed prime p and dimension d, there exists an integer n satisfying κ(p)+1 ≤ n ≤ p^d−κ(p)−2 such that every Cayley graph on the elementary abelian group (\mathbb{Z}_p)^d of degree n has a complete core, where κ(2)=5, κ(3)=12, and κ(p)=2 for primes p≥5.

Background

The paper proves that every p-ary Cayley graph of degree less than κ(p), or of degree at least p^d−κ(p), has a complete core. It also constructs, for every prime p, non-complete p-ary Cayley graph cores at the boundary degrees κ(p) and |V(X)|−κ(p)−1, showing that the sufficient degree bounds in Theorem 1.3 are optimal.

Question 4.2 asks whether any degree strictly between these boundary regimes nevertheless has the universal complete-core property: namely, whether every Cayley graph of that fixed degree on (\mathbb{Z}_p)^d has a complete core. The authors specifically note that the question appears open for p=2 when d≥7. This question is unresolved in the paper.