Richard Montgomery

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.

Logarithmically small minors and topological minors

Mader proved that for every integer

Rainbow spanning trees in properly coloured complete graphs

t

Almost All Friendly Matrices Have Many Obstructions

there is a smallest real number

Embedding bounded degree spanning trees in random graphs

c(t)

On the decomposition threshold of a given graph

such that any graph with average degree at least

Sharp threshold for embedding combs and other spanning trees in random graphs

c(t)

Hamiltonicity in random graphs is born resilient

must contain a

Universality for bounded degree spanning trees in randomly perturbed graphs

K_t

Embedding rainbow trees with applications to graph labelling and decomposition

-minor. Fiorini, Joret, Theis and Wood conjectured that any graph with