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.