Uniqueness of maximal dominating cycles in 3‐regular graphs and of hamiltonian cycles in 4‐regular graphs

Abstract

Abstract We construct 3‐regular (cubic) graphs G that have a dominating cycle C such that no other cycle C 1 of G satisfies V(C) ⊆ V ( C 1 ). By a similar construction we obtain loopless 4‐regular graphs having precisely one hamiltonian cycle. The basis for these constructions are considerations on the uniqueness of a cycle decomposition compatible with a given eulerian trail in some eulerian graph.