Decomposing k-arc-strong tournaments into strong spanning subdigraphs

Abstract

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 where is an in-branching rooted at the vertex u i and 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.