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 Erdős 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. 1 arXiv:1202.6219v2 [math.CO] 10 May 2013It is not clear whether the lower bound on r in Theorem 1.1 is best possible. However, as discussed below, there are oriented graphs whose in-and outdegrees are all very close to 3n/8 but which do not contain even a single Hamilton cycle. Moreover, for r < (3n − 4)/8, it is not even known whether an r-regular oriented graph contains a single Hamilton cycle (this is related to a conjecture of Jackson, see the survey [36] for a more detailed discussion). Both these facts indicate that any improvement in the lower bound on r would be extremely difficult to obtain.Regular tournaments obviously exist only if n is odd, but we still obtain an interesting corollary in the even case. Suppose that G is a tournament on n vertices where n is even and which is as regular as possible, i.e. the in-and outdegrees differ by 1. Then Theorem 1.1 implies that G has a decomposition into edge-disjoint Hamilton paths. Indeed, add an extra vertex to G which sends an edge to all vertices of G whose indegree is below (n − 1)/2 and which receives an edge from all others. The resulting tournament G is regular, and a Hamilton decomposition of G clearly corresponds to a decomposition of G into Hamilton paths.The difficulty of Kelly's conjecture is illustrated by the fact that even the existence of two edge-disjoint Hamilton cycles in a regular tournament is not obvious. The first result in this direction was proved by Jackson [22], who showed that every regular tournament on at least 5 vertices contains a Hamilton cycle and a Hamilton path which are edge-disjoint. Zhang [48] then demonstrated the existence of two edge-disjoint Hamilton cycles. These results were improved by considering Hamilton cycles in oriented graphs of large in-and outdegree by Thomassen [46], Häggkvist [20], Häggkvist and Thomason [21] as well as Kelly, Kühn and Osthus [25]. Keevash, Kühn and Osthus [24] then showed that every sufficiently large oriented graph G on n vertices whose in-and outdegrees...

Abstract. A fundamental theorem of Wilson states that, for every graph F , every sufficiently large F -divisible clique has an F -decomposition. Here a graph G is F -divisible if e(F ) divides e(G) and the greatest common divisor of the degrees of F divides the greatest common divisor of the degrees of G, and G has an F -decomposition if the edges of G can be covered by edge-disjoint copies of F . We extend this result to graphs G which are allowed to be far from complete. In particular, together with a result of Dross, our results imply that every sufficiently large K3-divisible graph of minimum degree at least 9n/10 + o(n) has a K3-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large K3-divisible graph with minimum degree at least 3n/4 has a K3-decomposition. We also obtain the asymptotically correct minimum degree thresholds of 2n/3 + o(n) for the existence of a C4-decomposition, and of n/2 + o(n) for the existence of a C 2ℓ -decomposition, where ℓ ≥ 3. Our main contribution is a general 'iterative absorption' method which turns an approximate or fractional decomposition into an exact one. In particular, our results imply that in order to prove an asymptotic version of Nash-Williams' conjecture, it suffices to show that every K3-divisible graph with minimum degree at least 3n/4 + o(n) has an approximate K3-decomposition,

Let H be any graph. We determine up to an additive constant the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let δ(H, n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G) ≥ k contains a perfect H-packing. We show thatThe value of χ * (H) depends on the relative sizes of the colour classes in the optimal colourings of H and satisfies χ(H) − 1 < χ * (H) ≤ χ(H).

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F -packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F . It is unlikely that there is a characterization of all graphs G which contain a perfect F -packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F -packing.The Regularity lemma of Szemerédi and the Blow-up lemma of Komlós, Sárközy and Szemerédi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F -packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved.

Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [6, 23] the same result was proved for 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs for any integer k ≥ 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform 'quasi-random' hypergraphs.

We say that a 3-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. Also, let C 4 denote the 3-uniform hypergraph on 4 vertices which contains 2 edges. We prove that for every ε > 0 there is an n 0 such that for every n n 0 the following holds: Every 3-uniform hypergraph on n vertices whose minimum degree is at least n/4 + εn contains a Hamilton cycle. Moreover, it also contains a perfect C 4 -packing. Both these results are best possible up to the error term εn.

We show that for each η > 0 every digraph G of sufficiently large order n is Hamiltonian if its out-and indegree sequences d +This gives an approximate solution to a problem of Nash- Williams (1975) [22] concerning a digraph analogue of Chvátal's theorem. In fact, we prove the stronger result that such digraphs G are pancyclic.

No abstract