Isoperimetric characterization of fast-robber cop number

Determine whether every connected undirected d-regular abelian Cayley graph on n vertices satisfies c_{1,\infty}(\Gamma)=\Theta_d\bigl(n\,\iota_v(\Gamma)\bigr), where \iota_v(\Gamma) is the vertex-isoperimetric number of \Gamma.

Background

The paper establishes the lower bound c_{1,\infty}(\Gamma)\ge \iota_v(\Gamma)n/(4d) for connected graphs of maximum degree d, and therefore obtains the lower-bound direction for connected undirected d-regular abelian Cayley graphs.

For the upper-bound direction, the paper proves a quotient-sweeping estimate in terms of the algebraic parameter N_H, associated with the largest cyclic quotient obtained after factoring out the subgroup generated by involutory generators. The authors note that the desired upper bound would follow from a comparison N_H=\Omega_d(1/\iota_v(\Gamma)). Establishing the stated equivalence would relate the fast-robber cop number to the graph’s vertex-isoperimetric scale for fixed-degree abelian Cayley graphs.