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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
##### Keywords:
pancyclic arc; tournament
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##### References:
 [1] Alspach, Canad Math Bull 10 pp 283– (1967) · Zbl 0148.43602 · doi:10.4153/CMB-1967-028-6 [2] Camion, CR Acad Sci Paris 249 pp 2151– (1959) [3] J Combin Inform System Sci 19 pp 207– (1994) [4] Topics on Tournaments, Holt, Rinehart and Winston, New York, 1968. [5] Moon, J Combin Inform System Sci 19 pp 207– (1994) [6] Yao, Discrete Appl Math 99 pp 245– (2000)
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