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Pancyclic arcs and connectivity in tournaments. (English) Zbl 1060.05037
Let \(p(T)\) denote the number of pancyclic arcs in a tournament \(T\) and let \(h(T)\) denote the maximum number of pancyclic arcs in the same Hamiltonian cycle of \(T\). Let \(p_k(n)\) and \(h_k(n)\) denote the minimum values of \(p(T)\) and \(h(T)\), respectively, over all \(k\)-strong tournaments \(T\) with \(n\) nodes. J. W. Moon [J. Comb. Inf. Syst. Sci. 19, No. 3–4, 207–214 (1994; Zbl 0860.05039)] showed that \(h(T)\geq 3\), for any non-trivial strong tournament, and characterized the minimal tournaments. The author of the present paper shows that if \(k\geq 2\), then \(h_k(n)\geq 5\) and \(p_k(n)\geq 2k+ 3\); and he characterizes tournaments with \(h(T)= 4\) and those with \(p(T)= 4\) or \(5\). He also conjectures, among other things, that if \(k\geq 2\), then \(h_k(n)\geq 2k +1\) and \(p_k(n)\) tends linearly to infinity with \(n\).

MSC:
05C20 Directed graphs (digraphs), tournaments
05C75 Structural characterization of families of graphs
05C38 Paths and cycles
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