On the vertex-pancyclicity of hypertournaments

A kk-hypertournament HH on nn vertices, where 2≤k≤n2≤k≤n, is a pair H=(V,AH)H=(V,AH), where VV is the vertex set of HH and AHAH is a set of kk-tuples of vertices, called arcs, such that, for all subsets S⊆VS⊆V with |S|=k|S|=k, AHAH contains exactly one permutation of SS as an arc. Gutin and Yeo (1997) showed in [2] that any strong kk-hypertournament HH on nn vertices, where 3≤k≤n−23≤k≤n−2, is Hamiltonian, and posed the question as to whether the result could be extended to vertex-pancyclicity. As a response, Petrovic and Thomassen (2006) in [4] and Yang (2009) in [6] gave some sufficient conditions for a strong hypertournament to be vertex-pancyclic.In this paper, we prove that, if HH is a strong kk-hypertournament on nn vertices, where 3≤k≤n−23≤k≤n−2, then HH is vertex-pancyclic. This extends the aforementioned results and Moon’s theorem for tournaments. Furthermore, our result is best possible in the sense that the bound k≤n−2k≤n−2 is tight.