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

Fractional clique decompositions of dense graphs

For each $r\ge 4$, we show that any graph $G$ with minimum degree at least $(1-1/100r)|G|$ has a fractional $K_r$-decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional $K_r$-decomposition given by Dukes (for small $r$) and Barber, K\"uhn, Lo, Montgomery and Osthus (for large $r$), giving the first bound that is tight up to the constant multiple of $r$ (seen, for example, by considering Tur\'an graphs). In combination with work by Glock, K\"uhn, Lo, Montgomery and Osthus, this shows that, for any graph $F$ with chromatic number $\chi(F)\ge 4$, and any $\varepsilon>0$, any sufficiently large graph $G$ with minimum degree at least $(1-1/100\chi(F)+\varepsilon)|G|$ has, subject to some further simple necessary divisibility conditions, an (exact) $F$-decomposition.Comment: 15 pages, 1 figure, submitte

Structural Decompositions for Problems with Global Constraints

A wide range of problems can be modelled as constraint satisfaction problems (CSPs), that is, a set of constraints that must be satisfied simultaneously. Constraints can either be represented extensionally, by explicitly listing allowed combinations of values, or implicitly, by special-purpose algorithms provided by a solver. Such implicitly represented constraints, known as global constraints, are widely used; indeed, they are one of the key reasons for the success of constraint programming in solving real-world problems. In recent years, a variety of restrictions on the structure of CSP instances have been shown to yield tractable classes of CSPs. However, most such restrictions fail to guarantee tractability for CSPs with global constraints. We therefore study the applicability of structural restrictions to instances with such constraints. We show that when the number of solutions to a CSP instance is bounded in key parts of the problem, structural restrictions can be used to derive new tractable classes. Furthermore, we show that this result extends to combinations of instances drawn from known tractable classes, as well as to CSP instances where constraints assign costs to satisfying assignments.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s10601-015-9181-

The existence of designs via iterative absorption: hypergraph FF-designs for arbitrary FF

We solve the existence problem for $F$-designs for arbitrary $r$-uniform hypergraphs~$F$. This implies that given any $r$-uniform hypergraph~$F$, the trivially necessary divisibility conditions are sufficient to guarantee a decomposition of any sufficiently large complete $r$-uniform hypergraph into edge-disjoint copies of~$F$, which answers a question asked e.g.~by Keevash. The graph case $r=2$ was proved by Wilson in 1975 and forms one of the cornerstones of design theory. The case when~$F$ is complete corresponds to the existence of block designs, a problem going back to the 19th century, which was recently settled by Keevash. In particular, our argument provides a new proof of the existence of block designs, based on iterative absorption (which employs purely probabilistic and combinatorial methods). Our main result concerns decompositions of hypergraphs whose clique distribution fulfills certain regularity constraints. Our argument allows us to employ a `regularity boosting' process which frequently enables us to satisfy these constraints even if the clique distribution of the original hypergraph does not satisfy them. This enables us to go significantly beyond the setting of quasirandom hypergraphs considered by Keevash. In particular, we obtain a resilience version and a decomposition result for hypergraphs of large minimum degree.Comment: This version combines the two manuscripts `The existence of designs via iterative absorption' (arXiv:1611.06827v1) and the subsequent `Hypergraph F-designs for arbitrary F' (arXiv:1706.01800) into a single paper, which will appear in the Memoirs of the AM

Hypergraph matchings and designs

We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC

Edge-decompositions of graphs with high minimum degree

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 $K_3$-divisible graph of minimum degree at least $9n/10+o(n)$ has a $K_3$-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large $K_3$-divisible graph with minimum degree at least $3n/4$ has a $K_3$-decomposition. We also obtain the asymptotically correct minimum degree thresholds of $2n/3 +o(n)$ for the existence of a $C_4$-decomposition, and of $n/2+o(n)$ for the existence of a $C_{2\ell}$-decomposition, where $\ell\ge 3$. 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 $K_3$-divisible graph with minimum degree at least $3n/4+o(n)$ has an approximate $K_3$-decomposition,Comment: 41 pages. This version includes some minor corrections, updates and improvement