Approximate Hamilton decompositions of robustly expanding regular digraphs

We show that every sufficiently large r-regular digraph G which has linear degree and is a robust outexpander has an approximate decomposition into edge-disjoint Hamilton cycles, i.e. G contains a set of r-o(r) edge-disjoint Hamilton cycles. Here G is a robust outexpander if for every set S which is not too small and not too large, the `robust' outneighbourhood of S is a little larger than S. This generalises a result of K\"uhn, Osthus and Treglown on approximate Hamilton decompositions of dense regular oriented graphs. It also generalises a result of Frieze and Krivelevich on approximate Hamilton decompositions of quasirandom (di)graphs. In turn, our result is used as a tool by K\"uhn and Osthus to prove that any sufficiently large r-regular digraph G which has linear degree and is a robust outexpander even has a Hamilton decomposition.Comment: Final version, published in SIAM Journal Discrete Mathematics. 44 pages, 2 figure

Hamilton decompositions of regular tournaments

We show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each \eta>0 every regular tournament G of sufficiently large order n contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968. Our result also extends to almost regular tournaments.Comment: 38 pages, 2 figures. Added section sketching how we can extend our main result. To appear in the Proceedings of the LM

Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments

A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdos on packing Hamilton cycles in random tournaments. As corollaries to the main result, we also obtain several results on packing Hamilton cycles in undirected graphs, giving e.g. the best known result on a conjecture of Nash-Williams. We also apply our result to solve a problem on the domination ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust decomposition lemma' for application in subsequent paper

Hamilton decompositions of regular bipartite tournaments

A regular bipartite tournament is an orientation of a complete balanced bipartite graph $K_{2n,2n}$ where every vertex has its in- and outdegree both equal to $n$. In 1981, Jackson conjectured that any regular bipartite tournament can be decomposed into Hamilton cycles. We prove this conjecture for all sufficiently large bipartite tournaments. Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs.Comment: 119 pages, 4 figure

Path and cycle decompositions of graphs and digraphs

In this thesis, we make progress on five long standing conjectures on path and cycle decompositions of graphs and digraphs. Firstly, we confirm a conjecture of Jackson from 1981 by showing that the edges of any sufficiently large regular bipartite tournament can be decomposed into Hamilton cycles. Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs. Secondly, we determine the minimum number of paths required to decompose the edges of any sufficiently large tournament of even order, thus resolving a conjecture of Alspach, Mason, and Pullman from 1976. We also prove an asymptotically optimal result for tournaments of odd order. Finally, we give asymptotically best possible upper bounds on the minimum number of paths, cycles, and cycles and edges required to decompose the edges of any sufficiently large dense graph. This makes progress on three famous conjectures from the 1960s: Gallai's conjecture, Hajós' conjecture, and the Erdős-Gallai conjecture, respectively. This includes joint work with António Girão, Daniela Kühn, Allan Lo, and Deryk Osthus

Cycle decompositions in k-uniform hypergraphs

We show that k-uniform hypergraphs on n vertices whose codegree is at least (2/3+o(1))n can be decomposed into tight cycles, subject to the trivial divisibility conditions. As a corollary, we show those graphs contain tight Euler tours as well. In passing, we also investigate decompositions into tight paths.In addition, we also prove an alternative condition for building absorbers for edge-decompositions of arbitrary k-uniform hypergraphs, which should be of independent interest

Cycle decompositions in k-uniform hypergraphs

We show that k-uniform hypergraphs on n vertices whose codegree is at least (2/3+o(1))n can be decomposed into tight cycles, subject to the trivial divisibility conditions. As a corollary, we show those graphs contain tight Euler tours as well. In passing, we also investigate decompositions into tight paths.In addition, we also prove an alternative condition for building absorbers for edge-decompositions of arbitrary k-uniform hypergraphs, which should be of independent interest

Decomposing tournaments into paths

We consider a generalisation of Kelly's conjecture which is due to 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 asks for the minimum number of paths needed in a path decomposition of a general tournament T . There is a natural lower bound for this number in terms of the degree sequence of T and it is conjectured that this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them