Decomposing tournaments into paths

Allan Lo,Viresh Patel,Jozef Skokan,John Talbot

doi:10.1016/j.endm.2017.07.040

Abstract

In this work we consider a generalisation of Kelly's conjecture which is due Alspach, Mason, and Pullman from 1976. Kelly's conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by Kühn and Osthus for large tournaments. The conjecture of Alspach, Mason, and Pullman concerns general tournaments and asks for the minimum number of paths needed in an edge decomposition of each tournament into paths. There is a natural lower bound for this number in terms of the degree sequence of the tournament and they conjecture this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them.

Highlights

There has been a great deal of recent activity in the study of decompositions of graphs and hypergraphs
The general prototypical question is this area asks whether, for some given class C of graphs, hypergraphs or directed graphs, the edge set of each H ∈ C can be decomposed into parts satisfying some given property
The special case of this problem where one wishes to establish the existence of Steiner systems asks for a decomposition of the edge set of the complete r-uniform hypergraph into r-uniform cliques of a fixed given size

Summary

Introduction

There has been a great deal of recent activity in the study of decompositions of graphs and hypergraphs. The special case of this problem where one wishes to establish the existence of Steiner systems asks for a decomposition of the edge set of the complete r-uniform hypergraph into r-uniform cliques of a fixed given size. Let D be a directed graph with vertex set V(D) and edge set E(D).

Results

Conclusion