Connected vertex-transitive digraphs on n vertices can have perimeter gap ≥ n/12, and every such digraph contains a directed cycle of length Ω(√n).
Hamiltonian cycles and paths in Cayley graphs and digraphs—a survey
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Long Directed Cycles in Vertex-Transitive Digraphs
Connected vertex-transitive digraphs on n vertices can have perimeter gap ≥ n/12, and every such digraph contains a directed cycle of length Ω(√n).