Pancyclic arcs and connectivity in tournaments

Description

A tournament is an orientation of the edges of a complete graph. An arc is pancyclic in a digraph D if it is contained in a cycle of length l, for every $3\leq l\leq |D|$. In [4], Moon showed that every strong tournament contains at least three pancyclics arcs and characterized the tournaments with exactly three pancyclic arcs. All these tournaments are not 2-strong. In this paper, we are interested in the minimum number $p_k(n)$ of pancyclic arcs in a k-strong tournament of order n. We conjecture that (for $k\geq 2$) there exists a constant $\alpha_k>0$ such that $p_k(n)\geq \alpha_kn$. After proving that every 2-strong tournament has a hamiltonian cycle containing at least five pancyclic arcs, we deduce that for $k\geq 2$, $p_k(n)\geq 2k+3$. We then characterize the tournaments having exactly four pancyclic arcs and those having exactly five pancyclic arcs.

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