Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

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 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: applications

In a recent paper, we showed 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. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following: (i) an undirected analogue of our result on robust outexpanders; (ii) best possible bounds on the size of an optimal packing of edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range of values for d. (iii) a similar result for digraphs of given minimum semidegree; (iv) an approximate version of a conjecture of Nash-Williams on Hamilton decompositions of dense regular graphs; (v) the observation that dense quasi-random graphs are robust outexpanders; (vi) a verification of the `very dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the size of an optimal packing of edge-disjoint Hamilton cycles in a random tournament.Comment: final version, to appear in J. Combinatorial Theory

Proof of the 1-factorization and Hamilton decomposition conjectures III: approximate decompositions

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $\chi'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint Hamilton cycles in a graph of given minimum degree. According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer questions of Nash-Williams from 1970. The above bounds are best possible. In the current paper, we show the following: suppose that $G$ is close to a complete balanced bipartite graph or to the union of two cliques of equal size. If we are given a suitable set of path systems which cover a set of `exceptional' vertices and edges of $G$, then we can extend these path systems into an approximate decomposition of $G$ into Hamilton cycles (or perfect matchings if appropriate).Comment: We originally split the proof into four papers, of which this was the third paper. We have now combined this series into a single publication [arXiv:1401.4159v2], which will appear in the Memoirs of the AMS. 29 pages, 2 figure

Decompositions of complete uniform hypergraphs into Hamilton Berge cycles

In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if $n$ divides $\binom{n}{k}$, then the complete $k$-uniform hypergraph on $n$ vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an alternating sequence $v_1,e_1,v_2,\dots,v_n,e_n$ of distinct vertices $v_i$ and distinct edges $e_i$ so that each $e_i$ contains $v_i$ and $v_{i+1}$. So the divisibility condition is clearly necessary. In this note, we prove that the conjecture holds whenever $k \ge 4$ and $n \ge 30$. Our argument is based on the Kruskal-Katona theorem. The case when $k=3$ was already solved by Verrall, building on results of Bermond

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