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Vertex-disjoint subtournaments of prescribed minimum outdegree or minimum semidegree: proof for tournaments of a conjecture of Stiebitz. (English) Zbl 1236.05095
Summary: It was proved S. Bessy, N. Lichiardopol and J.-S. Sereni [“Two proofs of the Bermond-Thomassen conjecture for tournaments with bounded minimum in-degree”, Discrete Math. 310, No. 3, 557–560 (2010; Zbl 1188.05072)] that for $$r \geq 1$$, a tournament with minimum semidegree at least $$2r - 1$$ contains at least $$r$$ vertex-disjoint directed triangles. It was also proved N. Lichiardopol [Discrete Math. 310, No. 19, 2567–2570 (2010; Zbl 1227.05157)] that for integers $$q \geq 3$$ and $$r \geq 1$$, every tournament with minimum semidegree at least $$(q - 1)r - 1$$ contains at least $$r$$ vertex-disjoint directed cycles of length $$q$$. None information was given on these directed cycles.
In this paper, we fill this gap a little. Namely, we prove that for $$d \geq 1$$ and $$r \geq 1$$, every tournament of minimum outdegree at least $$((d^2 + 3d + 2)/2)r - (d^2 + d + 2)/2$$ contains at least $$r$$ vertex-disjoint strongly connected subtournaments of minimum outdegree $$d$$. Next, we prove for tournaments a conjecture of Stiebitz stating that for integers $$s \geq 1$$ and $$t \geq 1$$, there exists a least number $$f(s, t)$$ such that every digraph with minimum outdegree at least $$f(s, t)$$ can be vertex-partitioned into two sets inducing subdigraphs with minimum outdegree at least $$s$$ and at least $$t$$, respectively.
Similar results related to the semidegree will be given. All these results are consequences of two results concerning the maximum order of a tournament of minimum outdegree $$d$$ (of minimum semidegree $$d$$) not containing proper subtournaments of minimum outdegree $$d$$ (of minimum semidegree $$d$$).

##### MSC:
 05C20 Directed graphs (digraphs), tournaments
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##### References:
 [1] J. Bang-Jensen and G. Gutin, Digraphs, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2001. · Zbl 0958.05002 [2] S. Bessy, N. Lichiardopol, and J.-S. Sereni, “Two proofs of the Bermond-Thomassen conjecture for tournaments with bounded minimum in-degree,” Discrete Mathematics, vol. 310, no. 3, pp. 557-560, 2010. · Zbl 1188.05072 · doi:10.1016/j.disc.2009.03.039 [3] N. Lichiardopol, “Vertex-disjoint directed cycles of prescribed length in tournaments with given minimum out-degree and in-degree,” Discrete Mathematics, vol. 310, no. 19, pp. 2567-2570, 2010. · Zbl 1227.05157 · doi:10.1016/j.disc.2010.06.024 [4] M. Stiebitz, “Decompositions of graphs and digraphs,” KAM Series in Discrete Mathematics-Combinatorics-Operation Research 95-309, 56-59. [5] N. Alon, “Splitting digraphs,” Combinatorics, Probability and Computing, vol. 15, no. 6, pp. 933-937, 2006. · Zbl 1116.05033 · doi:10.1017/S0963548306008042 [6] K. B. Reid, “Two complementary circuits in two-connected tournaments,” in Cycles in Graphs (Burnaby, B.C., 1982), vol. 115 of North-Holland Math. Stud., pp. 321-334, North-Holland, Amsterdam, The Netherlands, 1985. · Zbl 0573.05031 · doi:10.1016/S0304-0208(08)73025-1
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