Decomposing k-ARc-Strong Tournaments Into Strong Spanning Subdigraphs

• Authors: Jørgen Bang-Jensen;Anders Yeo
• Venue: Combinatorica
• Year: 2004

The so-called Kelly conjecture1 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.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 H1, H2, . . . , Hk such that each Hi is strongly connected.The last result implies that if T is a 74k-arc-strong tournament with speci.ed not necessarily distinct vertices u1, u2, . . . , uk, v1, v2, . . . , vk 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 ui and $$F^{ + }_{{v_{i} }}$$ is an out-branching rooted at the vertex vi, i=1,2, . . . , k. This solves a conjecture of Bang-Jensen and Gutin [3].We also discuss related problems and conjectures.