# Decomposing k-ARc-Strong Tournaments Into Strong Spanning Subdigraphs

The so-called Kelly conjectureFootnote 1 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.

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## Notes

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.

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Correspondence to Jørgen Bang-Jensen.

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Bang-Jensen, J., Yeo, A. Decomposing k-ARc-Strong Tournaments Into Strong Spanning Subdigraphs. Combinatorica 24, 331–349 (2004). https://doi.org/10.1007/s00493-004-0021-z

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