Edge-resolved shifts and abelian rigidity in cores of cubelike graphs

Abstract

We study cores of cubelike graphs through edge-resolved shifts. We prove three linked structural results. First, every prescribed edge of a core of a cubelike graph is reversed by a shift that maps every vertex to a neighbour. Second, for a connected normal Cayley graph on a finite abelian group, an edge {0,s} has such a witness exactly when there is an automorphism phi preserving the connection set with phi(s) = -s and s + Im(phi - id) contained in the connection set. Third, either of two explicit conditions—absence of a coset of a nonzero subgroup in the connection set, or a coherent family of witnesses with common defect image—forces the Cayley group to be elementary abelian, and hence the graph to be cubelike. These results isolate a local edge-to-defect-image mechanism and establish cubelikeness in two rigidity regimes; the unrestricted cubelike-core problem is outside the scope of this record.