A tight Erd\H{o}s-P\'osa function for wheel minors

Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t \geq 3$ there is a constant $c=c(t)$ such that for every integer $k\geq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor, or there is a subset $X$ of at most $c k \log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We conjecture that the result remains true more generally if we replace $W_t$ with any fixed planar graph $H$.Comment: 15 pages, 1 figur

Uniform Hyperbolicity of the Graphs of Curves

Let $\mathcal{C}(S_{g,p})$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur-Minksy and subsequently Bowditch showed that $\mathcal{C}(S_{g,p})$ is $\delta$-hyperbolic for some $\delta=\delta(g,p)$. In this paper, we show that there exists some $\delta>0$ independent of $g,p$ such that the curve graph $\mathcal{C}_{1}(S_{g,p})$ is $\delta$-hyperbolic. Furthermore, we use the main tool in the proof of this theorem to show uniform boundedness of two other quantities which a priori grow with $g$ and $p$: the curve complex distance between two vertex cycles of the same train track, and the Lipschitz constants of the map from Teichm\"{u}ller space to $\mathcal{C}(S)$ sending a Riemann surface to the curve(s) of shortest extremal length.Comment: 19 pages, 2 figures. This is a second version, revised to fix minor typos and to make the end of the main proof more understandabl

Small Complete Minors Above the Extremal Edge Density

A fundamental result of Mader from 1972 asserts that a graph of high average degree contains a highly connected subgraph with roughly the same average degree. We prove a lemma showing that one can strengthen Mader's result by replacing the notion of high connectivity by the notion of vertex expansion. Another well known result in graph theory states that for every integer t there is a smallest real c(t) so that every n-vertex graph with c(t)n edges contains a K_t-minor. Fiorini, Joret, Theis and Wood conjectured that if an n-vertex graph G has (c(t)+\epsilon)n edges then G contains a K_t-minor of order at most C(\epsilon)log n. We use our extension of Mader's theorem to prove that such a graph G must contain a K_t-minor of order at most C(\epsilon)log n loglog n. Known constructions of graphs with high girth show that this result is tight up to the loglog n factor

Logarithmically-small Minors and Topological Minors

Mader proved that for every integer $t$ there is a smallest real number $c(t)$ such that any graph with average degree at least $c(t)$ must contain a $K_t$-minor. Fiorini, Joret, Theis and Wood conjectured that any graph with $n$ vertices and average degree at least $c(t)+\epsilon$ must contain a $K_t$-minor consisting of at most $C(\epsilon,t)\log n$ vertices. Shapira and Sudakov subsequently proved that such a graph contains a $K_t$-minor consisting of at most $C(\epsilon,t)\log n \log\log n$ vertices. Here we build on their method using graph expansion to remove the $\log\log n$ factor and prove the conjecture. Mader also proved that for every integer $t$ there is a smallest real number $s(t)$ such that any graph with average degree larger than $s(t)$ must contain a $K_t$-topological minor. We prove that, for sufficiently large $t$, graphs with average degree at least $(1+\epsilon)s(t)$ contain a $K_t$-topological minor consisting of at most $C(\epsilon,t)\log n$ vertices. Finally, we show that, for sufficiently large $t$, graphs with average degree at least $(1+\epsilon)c(t)$ contain either a $K_t$-minor consisting of at most $C(\epsilon,t)$ vertices or a $K_t$-topological minor consisting of at most $C(\epsilon,t)\log n$ vertices.Comment: 19 page

Rainbow Tur\'an number of clique subdivisions

We show that for any integer $t\geq 2$, every properly edge-coloured graph on $n$ vertices with more than $n^{1+o(1)}$ edges contains a rainbow subdivision of $K_t$. Note that this bound on the number of edges is sharp up to the $o(1)$ error term. This is a rainbow analogue of some classical results on clique subdivisions and extends some results on rainbow Tur\'an numbers. Our method relies on the framework introduced by Sudakov and Tomon[2020] which we adapt to find robust expanders in the coloured setting.Comment:

Hitting and Harvesting Pumpkins

The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges. A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of G, each inducing a connected subgraph of G, such that there are at least c edges in G between A and B. We focus on covering and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time 2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be covered by at most k vertices. This generalizes known single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c=1,2 respectively. On the other hand, we present a O(log n)-approximation algorithm for both the problems of covering all c-pumpkin-models with a smallest number of vertices, and packing a maximum number of vertex-disjoint c-pumpkin-models.Comment: v2: several minor change

Automorphisms of the k-Curve Graph

Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that the automorphism group of the k-curve graph of a surface S is isomorphic to the extended mapping class group for all k sufficiently small with respect to the Euler characteristic of S. We prove the same result for the so-called systolic complex, a variant of the curve graph whose complete subgraphs encode the intersection patterns for any collection of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller.Comment: 22 pages, 11 figures, 1 tabl

Sublinear expanders and their applications

In this survey we aim to give a comprehensive overview of results using sublinear expanders. The term sublinear expanders refers to a variety of definitions of expanders, which typically are defined to be graphs $G$ such that every not-too-small and not-too-large set of vertices $U$ has neighbourhood of size at least $\alpha |U|$, where $\alpha$ is a function of $n$ and $|U|$. This is in contrast with linear expanders, where $\alpha$ is typically a constant. :We will briefly describe proof ideas of some of the results mentioned here, as well as related open problems.Comment: 39 pages, 15 figures. This survey will appear in `Surveys in Combinatorics 2024' (the proceedings of the 30th British Combinatorial Conference

Combinatorics and Probability

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. Both themes were richly represented at the workshop, with many recent exciting results presented by the lecturers