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Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths. (English) Zbl 1389.05058
For any integer \(k\), a tournament \(T\) is said to be strongly \(k\)-connected if the number of vertices of \(T\) is greater than \(k\) and the removal of any set of fewer than \(k\) vertices results in a strongly connected tournament. The main purpose of the present paper is to show that there exists an integer \(f\) such that every strongly \(f\)-connected tournament \(T\) admits a partition of its vertex set into \(j\) vertex classes \(V_1, V_2,\ldots ,V_j\) such that, for all \(i\), the subtournament induced on the tournament \(T\) by the vertex set \(V_i\) is strongly \(k\)-connected. In addition, it is shown that for any integer \(j\), there exists an integer \(h\) such that every strongly \(h\)-connected tournament has a \(1\)-factor consisting of \(j\) vertex-disjoint cycles of prescribed lengths. The paper also gives an estimate on the maximum number of operations required in computing the integers \(f\) and \(h\).

MSC:
05C20 Directed graphs (digraphs), tournaments
05C35 Extremal problems in graph theory
05C40 Connectivity
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[1] S. Abbasi: The solution of the El-Zahar Problem, Ph.D. Thesis, Rutgers University 1998.
[2] P. Camion: Chemins et circuits hamiltoniens des graphes complets, C. R. Acad. Sci.Paris 249 (1959), 2151–2152. · Zbl 0092.15801
[3] G. Chen, R.J. Gould and H. Li: Partitioning vertices of a tournament into independentcycles, J. Combin. Theory B 83 (2001), 213–220. · Zbl 1028.05038 · doi:10.1006/jctb.2001.2048
[4] P. Hajnal: Partition of graphs with condition on the connectivity and minimumdegree, Combinatorica 3 (1983), 95–99. · Zbl 0529.05030 · doi:10.1007/BF02579344
[5] P. Keevash and B. Sudakov: Triangle packings and 1-factors in oriented graphs, J. Combin. Theory B 99 (2009), 709–727. · Zbl 1208.05038 · doi:10.1016/j.jctb.2008.12.004
[6] D. KÜuhn, J. Lapinskas, D. Osthus and V. Patel: Proof of a conjecture of Thomassen on Hamilton cycles in highly connected tournaments, Proc. London Math Soc. 109 (2014), 733–762. · Zbl 1302.05069 · doi:10.1112/plms/pdu019
[7] K. B. Reid: Two complementary circuits in two-connected tournaments, in: Cycles in Graphs, B.R. Alspach, C.D. Godsil, Eds., Ann. Discrete Math. 27 (1985), 321–334. · doi:10.1016/S0304-0208(08)73025-1
[8] K. B. Reid: Three problems on tournaments, Graph Theory and its applications:East and West, Ann. New York Acad. Sci. 576 (1989), 466–473.
[9] Z. M. Song: Complementary cycles of all lengths in tournaments, J. Combin. Theory B 57 (1993), 18–25. · Zbl 0723.05062 · doi:10.1006/jctb.1993.1002
[10] C. Thomassen: Graph decomposition with constraints on the connectivity and minimumdegree, J. Graph Theory 7 (1983), 165–167. · Zbl 0515.05045 · doi:10.1002/jgt.3190070204
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