Clique minors in graphs with a forbidden subgraph

The classical Hadwiger conjecture dating back to 1940's states that any graph of chromatic number at least $r$ has the clique of order $r$ as a minor. Hadwiger's conjecture is an example of a well studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on $n$ vertices of independence number $\alpha(G)$ at most $r$. If true Hadwiger's conjecture would imply the existence of a clique minor of order $n/\alpha(G)$. Results of Kuhn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that $G$ is $H$-free for some bipartite graph $H$ then one can find a polynomially larger clique minor. This has recently been extended to triangle free graphs by Dvo\v{r}\'ak and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph $H$, answering a question of Dvo\v{r}\'ak and Yepremyan. In particular, we show that any $K_s$-free graph has a clique minor of order $c_s(n/\alpha(G))^{1+\frac{1}{10(s-2) }}$, for some constant $c_s$ depending only on $s$. The exponent in this result is tight up to a constant factor in front of the $\frac{1}{s-2}$ term.Comment: 11 pages, 1 figur

Hypergraph matchings and designs

We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC

The robust component structure of dense regular graphs and applications

In this paper, we study the large-scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a graph is robustly expanding if it still expands after the deletion of a small fraction of its vertices and edges. Our main result allows us to harness the useful consequences of robust expansion even if the graph itself is not a robust expander. It states that every dense regular graph can be partitioned into `robust components', each of which is a robust expander or a bipartite robust expander. We apply our result to obtain (amongst others) the following. (i) We prove that whenever \eps >0, every sufficiently large 3-connected D-regular graph on n vertices with D \geq (1/4 + \eps)n is Hamiltonian. This asymptotically confirms the only remaining case of a conjecture raised independently by Bollob\'as and H\"aggkvist in the 1970s. (ii) We prove an asymptotically best possible result on the circumference of dense regular graphs of given connectivity. The 2-connected case of this was conjectured by Bondy and proved by Wei.Comment: final version, to appear in the Proceedings of the LMS. 36 pages, 1 figur