Long Directed Cycles in Vertex-Transitive Digraphs

The search for Hamiltonian cycles in vertex-transitive graphs and digraphs is a classical problem at the interface of graph theory and group theory. In the undirected setting, this goes back to famous conjectures of Lov\'asz and Thomassen predicting that every sufficiently large connected vertex-transitive graph is Hamiltonian. The directed analogue has an even richer history, originating with Rankin in 1946, naturally translating the search for long cycles into classical group rearrangement problems. It was shown by Trotter and Erd\H{o}s in 1978 that connected vertex-transitive digraphs need not be Hamiltonian. In 1981, Alspach asked whether there exist connected vertex-transitive digraphs whose longest directed cycle misses arbitrarily many vertices. This question was only recently resolved by Buci\'c, Hendrey, Mohar, Steiner and Yepremyan, who constructed connected vertex-transitive digraphs on $n$ vertices whose longest directed cycle omits $(1-o(1))\log n$ vertices. They further conjectured that the number of omitted vertices can grow linearly with $n$, remarking that it would already be interesting to improve their logarithmic lower bound to a polynomial bound. In this paper, we confirm their conjecture in a strong form by constructing infinitely many connected vertex-transitive digraphs on $n$ vertices whose longest directed cycle omits at least $n/12$ vertices. In the same work, Buci\'c, Hendrey, Mohar, Steiner and Yepremyan also proved that every connected vertex-transitive digraph on $n$ vertices contains a directed cycle of length $\Omega(n^{1/3})$, giving the first lower bound for this problem that grows with $n$. We improve this to $\Omega(\sqrt n)$, matching the order of Babai's classical theorem from 1979 for undirected vertex-transitive graphs.