From One to Many Rainbow Hamiltonian Cycles

Abstract

Given a graph G and a family \(\mathcal {G} = \{G_1,\ldots ,G_n\}\) of subgraphs of G, a transversal of \(\mathcal {G}\) is a pair \((T,\phi )\) such that \(T \subseteq E(G)\) and \(\phi : T \rightarrow [n]\) is a bijection satisfying \(e \in G_{\phi (e)}\) for each \(e \in T\). We call a transversal \((T, \phi )\) 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 \(G_i \in \mathcal {G}\), for every \(\mathcal {G}\) which contains a Hamiltonian transversal, the number of Hamiltonian transversals contained in \(\mathcal {G}\) is bounded below by a function of G’s maximum degree. This generalizes a theorem of Thomassen stating that, for \(m \ge 300\), no m-regular graph is uniquely Hamiltonian. We also extend Joos and Kim’s recent result that, if \(G = K_{n}\) and each \(G_i \in \mathcal {G}\) has minimum degree at least n/2, then \(\mathcal {G}\) has a Hamiltonian transversal: we show that, in this setting, \(\mathcal {G}\) has factorially many Hamiltonian transversals. Finally, we prove analogues of both of these theorems for transversals which form perfect matchings in G.