Andrew Treglown

Matchings in 3-uniform hypergraphs

We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph. More precisely, suppose that H is a sufficiently large 3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex degree of H is greater than (n-12)-(2n/32), then H contains a perfect matching. This bound is tight and answers a question of Han, Person and Schacht. More generally, we show that H contains a matching of size d=

Exact minimum degree thresholds for perfect matchings in uniform hypergraphs

Abstract Given positive integers k and l where 4 divides k and k / 2 ⩽ l ⩽ k − 1 , we give a minimum l-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, as well as the hypergraph removal lemma and a structural result of Keevash and Sudakov relating to the Turan number of the expanded triangle.

Proof of the 1-factorization and Hamilton decomposition conjectures

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 ≥ 2⌈n/4⌉ − 1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ′(G) = D. (ii) [Hamilton decomposition conjecture] Suppose that D ≥ ⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that G is a graph on n vertices with minimum degree δ ≥ n/2. Then G contains at least (n − 2)/8 edge-disjoint Hamilton cycles. According to Dirac, (i) was first raised in the 1950’s. (ii) and (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.

HAMILTON DECOMPOSITIONS OF REGULAR TOURNAMENTS

We show that every sufficiently large regular tournament can a lmost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each � > 0 every regular tournament G of sufficiently large ordern contains at least (1/2 �)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.

An Ore-type Theorem for Perfect Packings in Graphs

We say that a graph

The number of maximal sum-free subsets of integers

G

A random version of Sperner's theorem

has a perfect

Embedding spanning bipartite graphs of small bandwidth

H

A degree sequence Hajnal-Szemerédi theorem

-packing (also called an

A note on some embedding problems for oriented graphs

H