From one to many rainbow Hamiltonian cycles

Abstract

Given a graph G and a family G = {G 1 , . . . , Gn} of subgraphs of G, a transversal of G is a pair (T, φ) such that T ⊆ E(G) and φ : T → [n] is a bijection satisfying e ∈ G φ(e) for each e ∈ T . We call a transversal Hamiltonian if T corresponds to the edge set of a Hamiltonian cycle in G. We show that, under certain conditions on the maximum degree of G and the minimum degrees of the G i ∈ G, for every G which contains a Hamiltonian transversal, the number of Hamiltonian transversals contained in G is bounded below by a function of G's maximum degree. This generalizes a theorem of Thomassen stating that, for m ≥ 300, no m-regular graph is uniquely Hamiltonian. We also extend Joos and Kim's recent result that, if G = Kn and each G i ∈ G has minimum degree at least n 2 , then G has a Hamiltonian transversal: we show that, in this setting, G has exponentially many Hamiltonian transversals. Finally, we prove analogues of both of these theorems for transversals which form perfect matchings of G.