Small cutsets in quasiminimal Cayley graphs

AbstractWe continue the recent study carried out by several authors on the cut sets in Cayley graphs with respect to quasiminimal generating sets. We improve the known results on these questions.The application of our main theorem to symmetric Cayley graphs on minimal generating sets leads to the following result.Let G be a group containing a minimal generating set M such that | M | ⩾ 4. Let S = M ∪ M−1. Then one of the following conditions holds. 1.(i) s2 = u2 and u4 = 1, for all s, u ∈ M2.(ii) For all (d + 1)-subsets A and B of G which are not of the form Γ (x) ∪ {x} for any x ∈ G, there exists d + 1 disjoint paths from A to B in Cay(G, S)