Julia Böttcher

Properly coloured copies and rainbow copies of large graphs with small maximum degree

Let G be a graph on n vertices with maximum degree Δ. We use the Lovasz local lemma to show the following two results about colourings χ of the edges of the complete graph Kn. If for each vertex v of Kn the colouring χ assigns each colour to at most (n - 2)/(22.4Δ2) edges emanating from v, then there is a copy of G in Kn which is properly edge-coloured by χ. This improves on a result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4), 409–433, 2003]. On the other hand, if χ assigns each colour to at most n/(51Δ2) edges of Kn, then there is a copy of G in Kn such that each edge of G receives a different colour from χ. This proves a conjecture of Frieze and Krivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on a framework developed by Lu and Szekely [Electron. J. Comb. 14(1), R63, 2007] for applying the local lemma to random injections. In order to improve the constants in our results we use a version of the local lemma due to Bissacot, Fernandez, Procacci, and Scoppola [preprint, arXiv:0910.1824].

AN APPROXIMATE VERSION OF THE TREE PACKING CONJECTURE

We prove that for any pair of constants ɛ > 0 and Δ and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most Δ, and with at most (n2) edges in total packs into

AN EXTENSION OF THE BLOW-UP LEMMA TO ARRANGEABLE GRAPHS ∗

{K_{(1 + \varepsilon )n}}

Tight cycles and regular slices in dense hypergraphs

. This implies asymptotic versions of the Tree Packing Conjecture of Gyárfás from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

Embedding spanning bounded degree subgraphs in randomly perturbed graphs

A conjecture by Bollobas and Komlos states the following: For every@c>0and integersr>=2and @D, there exists@b>0with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least(r-1r+@c)nand H is an r-chromatic graph with n vertices, bandwidth at most @bn and maximum degree at most @D, then G contains a copy of H. This conjecture generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs. We prove the conjecture for the case r=3.

Spanning embeddings of arrangeable graphs with sublinear bandwidth

The blow-up lemma established by Komlos, Sarkozy, and Szemeredi in 1997 is an important tool for the embedding of spanning subgraphs of bounded maximum degree. Here we prove several generalizations of this result concerning the embedding of

Turánnical hypergraphs

a

Embedding into Bipartite Graphs

-arrangeable graphs, where a graph is called