Decomposing k-ARc-Strong Tournaments Into Strong Spanning Subdigraphs

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.