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Combinatorica

, Volume 24, Issue 3, pp 331–349 | Cite as

Decomposing k-ARc-Strong Tournaments Into Strong Spanning Subdigraphs

  • Jørgen Bang-Jensen
  • Anders Yeo
Original Paper

The so-called Kelly conjecture1 states that every regular tournament on 2k+1 vertices has a decomposition into k-arc-disjoint hamiltonian cycles. In this paper we formulate a generalization of that conjecture, namely we conjecture that every k-arc-strong tournament contains k arc-disjoint spanning strong subdigraphs. We prove several results which support the conjecture:

If D = (V, A) is a 2-arc-strong semicomplete digraph then it contains 2 arc-disjoint spanning strong subdigraphs except for one digraph on 4 vertices.

Every tournament which has a non-trivial cut (both sides containing at least 2 vertices) with precisely k arcs in one direction contains k arc-disjoint spanning strong subdigraphs. In fact this result holds even for semicomplete digraphs with one exception on 4 vertices.

Every k-arc-strong tournament with minimum in- and out-degree at least 37k contains k arc-disjoint spanning subdigraphs H 1, H 2, . . . , H k such that each H i is strongly connected.

The last result implies that if T is a 74k-arc-strong tournament with speci.ed not necessarily distinct vertices u 1, u 2, . . . , u k , v 1, v 2, . . . , v k then T contains 2k arc-disjoint branchings \( F^{ - }_{{u_{1} }} ,F^{ - }_{{u_{2} }} ,...,F^{ - }_{{u_{k} }} ,F^{ + }_{{v_{1} }} ,F^{ + }_{{v_{2} }} ,...,F^{ + }_{{v_{k} }} \) where \( F^{ - }_{{u_{i} }} \) is an in-branching rooted at the vertex u i and \( F^{ + }_{{v_{i} }} \) is an out-branching rooted at the vertex v i , i=1,2, . . . , k. This solves a conjecture of Bang-Jensen and Gutin [3].

We also discuss related problems and conjectures.

Footnotes
  1. 1.

    1 A proof of the Kelly conjecture for large k has been announced by R. Häggkvist at several conferences and in [5] but to this date no proof has been published.

Mathematics Subject Classification (2000):

05C20 05C38 05C40 05C70 

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Copyright information

© János Bolyai Mathematical Society 2004

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science University of Southern DenmarkOdenseDenmark
  2. 2.Department of Computer ScienceRoyal Holloway University of LondonEgham Surrey TW20 0EXUnited Kingdom

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