Allan Lo

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.

Fractional Clique Decompositions of Dense Graphs and Hypergraphs

Abstract Our main result is that every graph G on n ≥ 10 4 r 3 vertices with minimum degree δ ( G ) ≥ ( 1 − 1 / 10 4 r 3 / 2 ) n has a fractional K r -decomposition. Combining this result with recent work of Barber, Kuhn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) F-decompositions for a wide class of graphs F (including large cliques). For general k-uniform hypergraphs, we give a short argument which shows that there exists a constant c k > 0 such that every k-uniform hypergraph G on n vertices with minimum codegree at least ( 1 − c k / r 2 k − 1 ) n has a fractional K r ( k ) -decomposition, where K r ( k ) is the complete k-uniform hypergraph on r vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilsons theorem that every large F-divisible complete graph has an F-decomposition.

Perfect matchings in 3-partite 3-uniform hypergraphs

Let H be a 3-partite 3-uniform hypergraph, i.e. a 3-uniform hypergraph such that every edge intersects every partition class in exactly one vertex, with each partition class of size n. We determine a Dirac-type vertex degree threshold for perfect matchings in 3-partite 3-uniform hypergraphs.

Properly coloured Hamiltonian cycles in edge-coloured complete graphs

Let Kcn be an edge-coloured complete graph on n vertices. Let Δmon(Kcn) denote the largest number of edges of the same colour incident with a vertex of Kcn. A properly coloured cycleis a cycle such that no two adjacent edges have the same colour. In 1976, BollobÁs and ErdŐs[6] conjectured that every Kcn with Δmon(Kcn)<⌊n/2⌋contains a properly coloured Hamiltonian cycle. In this paper, we show that for any ε>0, there exists an integer n0 such that every Kcn with Δmon(Kcn)<(1/2–ε)n and n≥n0 contains a properly coloured Hamiltonian cycle. This improves a result of Alon and Gutin [1]. Hence, the conjecture of BollobÁs and ErdŐs is true asymptotically.

An Edge-Colored Version of Dirac's Theorem

Let

A Dirac Type Condition for Properly Coloured Paths and Cycles

G

Clique decompositions of multipartite graphs and completion of latin squares

be an edge-colored graph. The minimum color degree

Cliques in graphs

\delta^c(G)

l-Degree Turan Density

of

A note on the minimum size of k-rainbow-connected graphs☆

G