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Cycle-pancyclism in tournaments. II. (English) Zbl 0844.05047

Summary: Let \(T\) be a Hamiltonian tournament with \(n\) vertices and \(\gamma\) a Hamiltonian cycle of \(T\). We develop a general method to find cycles of length \(k\) with \({n+ 4\over 2}< k< n\), intersecting \(\gamma\) in a large number of arcs. In particular we can show that if there does not exist a cycle \(C_k\) intersecting \(\gamma\) in at least \(k- 3\) arcs then for any arc \(e\) of \(\gamma\) there exists a cycle \(C_k\) containing \(e\) and intersecting \(\gamma\) in at least \(k- {2(n- 3)\over n- k+ 3}- 2\) arcs. In a previous paper [ibid. 11, No. 3, 233-243 (1995; Zbl 0833.05039)] the case of cycles of length \(k\) with \(k\leq {n+ 4\over 2}\) was studied.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
05C45 Eulerian and Hamiltonian graphs

Citations:

Zbl 0833.05039
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References:

[1] Alspach, B.: Cycles of each length in regular tournaments. Canad. Math. Bull.10, 283–286 (1967) · Zbl 0148.43602 · doi:10.4153/CMB-1967-028-6
[2] Bermond, J.C., Thomasen, C.: Cycles in digraphs – A survey. J. Graph Theory,5, 43, 145–157 (1981) · Zbl 0458.05035 · doi:10.1002/jgt.3190050102
[3] Galeana-Sanchez, H., Rajsbaum, S.: Cycle Pancyclism in Tournaments I, Pub. Prel. 266 (technical report), Instituto de Matemáticas, UNAM, Mexico, April 1992. Submitted for publication
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