Extremal graph theory via structural analysis

We discuss two extremal problems in extremal graph theory. First we establish a precise characterisation of 4-uniform hypergraphs with minimum codegree close to n/2 which contain a Hamilton 2-cycle. As a corollary we determine the exact Dirac threshold for Hamilton 2-cycles in 4-uniform hypergraphs, and we provide a polynomial-time algorithm which answers the corresponding decision problem for 4-graphs with minimum degree close to n/2. In contrast we also show that the corresponding decision problem for tight Hamilton cycles in dense k-graphs is NP-complete. Furthermore we study the following bootstrap percolation process: given a connected graph G, we infect an initial set A of vertices, and in each step a vertex v becomes infected if at least a p-proportion of its neighbours are infected. A set A which infects the whole graph is called a contagious set. Our main result states that for every pin (0,1] and for every connected graph G on n vertices the minimal size of a contagious set is less than 2pn or 1. This result is best-possible, but we provide a stronger bound in the case of graphs of girth at least five. Both proofs exploit the structure of a minimal counterexample

Euler tours in hypergraphs

We show that a quasirandom $k$-uniform hypergraph $G$ has a tight Euler tour subject to the necessary condition that $k$ divides all vertex degrees. The case when $G$ is complete confirms a conjecture of Chung, Diaconis and Graham from 1989 on the existence of universal cycles for the $k$-subsets of an $n$-set.Comment: version accepted for publication in Combinatoric

The hyperbolic geometry of random transpositions

Turn the set of permutations of $n$ objects into a graph $G_n$ by connecting two permutations that differ by one transposition, and let $\sigma_t$ be the simple random walk on this graph. In a previous paper, Berestycki and Durrett [In Discrete Random Walks (2005) 17--26] showed that the limiting behavior of the distance from the identity at time $cn/2$ has a phase transition at $c=1$. Here we investigate some consequences of this result for the geometry of $G_n$. Our first result can be interpreted as a breakdown for the Gromov hyperbolicity of the graph as seen by the random walk, which occurs at a critical radius equal to $n/4$. Let $T$ be a triangle formed by the origin and two points sampled independently from the hitting distribution on the sphere of radius $an$ for a constant $0<a<1$. Then when $a<1/4$, if the geodesics are suitably chosen, with high probability $T$ is $\delta$-thin for some $\delta>0$, whereas it is always O(n)-thick when $a>1/4$. We also show that the hitting distribution of the sphere of radius $an$ is asymptotically singular with respect to the uniform distribution. Finally, we prove that the critical behavior of this Gromov-like hyperbolicity constant persists if the two endpoints are sampled from the uniform measure on the sphere of radius $an$. However, in this case, the critical radius is $a=1-\log2$.Comment: Published at http://dx.doi.org/10.1214/009117906000000043 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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