Ewan Davies

For a graph $G$ and $p∈[0,1]$, we denote by $G_p$ the random sparsification of $G$ obtained by keeping each edge of $G$ independently, with probability $p$. We show that there exists a $C>0$ such that if $p≥C(\log n)^{1/3}n^{-2/3}$ and $G$ is an $n$-vertex graph with $n∈3\mathbbN$ and $δ(G)≥2n/3$, then with high probability $G_p$ contains a triangle factor. Both the minimum degree condition and the probability condition, up to the choice of $C$, are tight. Our result can be viewed as a common strengthening of the seminal theorems of Corrádi and Hajnal, which deals with the extremal minimum degree condition for containing triangle factors (corresponding to $p=1$ in our result), and Johansson, Kahn and Vu, which deals with the threshold for the appearance of a triangle factor in $G(n,p)$ (corresponding to $G=K_n$ in our result). It also implies a lower bound on the number of triangle factors in graphs with minimum degree at least $2n/3$ which gets close to the truth.

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets.

With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $∆$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2\le t\le ∆^\frac{2\varepsilon}{1+2\varepsilon}/(\log∆)^2$, with $⌊(1+\varepsilon)∆/\log(∆/\sqrtt)⌋$ colours.

This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor $2$ and it coincides with a famous algorithmic barrier to colouring random graphs.

The first author together with Jenssen, Perkins and Roberts (2017) recently showed how local properties of the hard-core model on triangle-free graphs guarantee the existence of large independent sets, of size matching the best-known asymptotics due to Shearer (1983). The present work strengthens this in two ways: first, by guaranteeing stronger graph structure in terms of colourings through applications of the Lovász local lemma; and second, by extending beyond triangle-free graphs in terms of local sparsity, treating for example graphs of bounded local edge density, of bounded local Hall ratio, and of bounded clique number. This generalises and improves upon much other earlier work, including that of Shearer (1995), Alon (1996) and Alon, Krivelevich and Sudakov (1999), and more recent results of Molloy (2019), Bernshteyn (2019) and Achlioptas, Iliopoulos and Sinclair (2019). Our results derive from a common framework built around the hard-core model. It pivots on a property we call local occupancy, giving a clean separation between the methods for deriving graph structure with probabilistic information and verifying the requisite probabilistic information itself.

We develop an approximation algorithm for the partition function of the ferromagnetic Potts model on graphs with a small-set expansion condition, and as a step in the argument we give a graph partitioning algorithm with expansion and minimum degree conditions on the subgraphs induced by each part. These results extend previous work of Jenssen, Keevash, and Perkins (2019) on the Potts model and related problems in expander graphs, and of Oveis Gharan and Trevisan (2014) on partitioning into expanders.

We give a new approach to handling hypergraph regularity. This approach allows for vertex-by-vertex embedding into regular partitions of hypergraphs, and generalises to regular partitions of sparse hypergraphs. We also prove a corresponding sparse hypergraph regularity lemma.

We give algorithms for approximating the partition function of the ferromagnetic Potts model on $d$-regular expanding graphs. We require much weaker expansion than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models, using extremal graph theory and applications of Karger’s algorithm to counting cuts that may be of independent interest. It is #BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of #BIS are rare. We believe that these methods can shed more light on other important problems such as sub-exponential algorithms for approximate counting problems.

The ferromagnetic Ising model is a model of a magnetic material and a central topic in statistical physics. It also plays a starring role in the algorithmic study of approximate counting: approximating the partition function of the ferromagnetic Ising model with uniform external field is tractable at all temperatures and on all graphs, due to the randomized algorithm of Jerrum and Sinclair.

Here we show that hidden inside the model are hard computational problems. For the class of bounded-degree graphs we find computational thresholds for the approximate counting and sampling problems for the ferromagnetic Ising model at fixed magnetization (that is, fixing the number of $+1$ and $-1$ spins).

In particular, letting $\beta_c(∆)$ denote the critical inverse temperature of the zero-field Ising model on the infinite $∆$-regular tree, and $\eta_{∆,β,1}^+$ denote the mean magnetization of the zero-field $+$ measure on the infinite $∆$-regular tree at inverse temperature $β$, we prove, for the class of graphs of maximum degree $∆$:

1. For $β< \beta_c(∆)$ there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for all magnetizations $η$.

2. For $β> \beta_c(∆)$, there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for magnetizations $η$ such that $|η| >\eta_{∆,β,1}^+$.

3. For $β> \beta_c(∆)$, there is no FPRAS for the fixed-magnetization Ising model for magnetizations $η$ such that $|η| <\eta_{∆,β,1}^+$ unless NP=RP.

List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a $k$-list-assignment $L$ of a graph $G$, which is the assignment of a list $L(v)$ of $k$ colours to each vertex $v∈V(G)$, we study the existence of $k$ pairwise-disjoint proper colourings of $G$ using colours from these lists. We may refer to this as a list-packing. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest $k$ for which such a list-packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of $G$. (The reader might already find it interesting that such a minimal $k$ is well defined.) We also pursue a more focused study of the case when $G$ is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal $k$ above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalisations of the problem above in the same spirit.

In 1967, Erdős asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erdős and Hajnal together with Shearer’s classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows that $f(n)$ is at most $(2 \sqrt{2} + o(1)) \sqrt{n/\log n}$. We improve this bound by a factor $\sqrt{2}$, as well as obtaining an analogous bound on the list chromatic number which is tight up to a constant factor. A bound in terms of the number of edges that is similarly tight follows, and these results confirm a conjecture of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot.

We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $\alpha_c(∆)$ and provide (i) for $α< \alpha_c(∆)$ randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most $αn$ in $n$-vertex graphs of maximum degree $∆$; and (ii) a proof that unless NP=RP, no such algorithms exist for $α>\alpha_c(∆)$. The critical density is the occupancy fraction of the hard core model on the clique $K_{∆+1}$ at the uniqueness threshold on the infinite $∆$-regular tree, giving $\alpha_c(∆)∼\frace{1+e}\frac{1}{∆}$ as $∆\to∞$.

We obtain an approximate sparse hypergraph version of the blow-up lemma, showing that partite hypergraphs with sufficient regularity of small subgraph counts behave as if they were complete partite for the purpose of embedding bounded degree hypergraphs.

We prove that the ‘Upper Matching Conjecture’ of Friedland, Krop, and Markström and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every $d$ and every large enough $n$ divisible by $2d$, a union of $n/(2d)$ copies of the complete $d$-regular bipartite graph maximizes the number of independent sets and matchings of size $k$ for each $k$ over all $d$-regular graphs on $n$ vertices. To prove this we utilize the cluster expansion for the canonical ensemble of a statistical physics spin model, and we give some further applications of this method to maximizing and minimizing the number of independent sets and matchings of a given size in regular graphs of a given minimum girth.

We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the “yes” case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the “yes” case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would be strong negative evidence for the truth of the Unique Games Conjecture.

Given $\varepsilon>0$, there exists $f_0$ such that, if $f_0 \le f \le ∆^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $∆$ in which the neighbourhood of every vertex in $G$ spans at most $∆^2/f$ edges, we have (i) an independent set drawn uniformly at random from $G$ has expected size at least $(1/2-\varepsilon)(n/∆)\log f$; and (ii) the fractional chromatic number of $G$ is at most $(2+\varepsilon)∆/\log f$.

The asymptotic factors cannot in general be improved by more than a factor of $2$. One may view these as stronger versions of results of Ajtai, Komlós and Szemerédi (1981) and Shearer (1983) on the independence number. The proofs of both use a tight analysis of the hard-core model.

We give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any $∆∈\mathbbN$ and any $k≥3.02 ∆+1$, there exists an open set $U$ in the complex plane that contains the interval $[0,1)$ such that $\mathbfZ(G;k,w)≠0$ for any $w∈U$ and any graph $G$ of maximum degree at most $∆$. For small values of $∆$ we are able to give better results. As an application of our results we obtain improved bounds on $k$ for the existence of deterministic approximation algorithms for counting the number of proper $k$-colourings of graphs of small maximum degree.

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow-up work of Bernshteyn) on the (list) chromatic number of triangle-free graphs. In both our results, we permit the amount of colour made available to vertices of lower degree to be accordingly lower. One result concerns list colouring and correspondence colouring, while the other concerns fractional colouring. Our proof of the second illustrates the use of the hard-core model to prove a Johansson-type result, which may be of independent interest.

We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$-state Potts model on $4$-regular graphs, for $q\ge 5$. This proves the first case of a conjecture of the author, Perkins, Jenssen, and Roberts on extensions of their methods, and implies tight bounds on the antiferromagnetic Potts partition function.

The zero-temperature limit gives upper and lower bounds on the number of proper $q$-colourings of $4$-regular graphs, which almost proves the case $d=4$ of a conjecture of Galvin and Tetali. For any $q \ge 5$ we prove that the number of proper $q$-colourings of a $4$-regular graph is maximised by a union of $K_{4,4}$’s.

Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hard-core model and monomer-dimer model are the independence and matching polynomials respectively.

We show how stability results follow naturally from the recently developed occupancy method for maximizing and minimizing physical observables over classes of regular graphs, and then show these stability results can be used to obtain tight extremal bounds on the individual coefficients of the corresponding partition functions.

As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of $q$-colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least $5$.

We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$. This immediately implies corresponding tight bounds on the anti-ferromagnetic Potts partition function.

Taking the zero-temperature limit gives new results in extremal combinatorics: the number of $q$-colorings of a $3$-regular graph, for any $q \ge 2$, is maximized by a union of $K_{3,3}$’s. This proves the $d=3$ case of a conjecture of Galvin and Tetali.

We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on $n$ vertices with maximum degree $d$, showing that an independent set drawn uniformly at random from such a graph has expected size at least $(1+o_d(1)) \frac{\log d}dn$. This gives an alternative proof of Shearer’s upper bound on the Ramsey number $R(3,k)$. We then prove that the total number of independent sets in a triangle-free graph with maximum degree $d$ is at least $\exp \left[\left(\frac{1}{2}+o_d(1) \right) \frac{\log^2 d}dn \right]$. The constant $1/2$ in the exponent is best possible. In both cases, tightness is exhibited by a random $d$-regular graph.

Both results come from considering the hard-core model from statistical physics: a random independent set $I$ drawn from a graph with probability proportional to $λ^{|I|}$, for a fugacity parameter $λ>0$. We prove a general lower bound on the occupancy fraction (normalized expected size of the random independent set) of the hard-core model on triangle-free graphs of maximum degree $d$. The bound is asymptotically tight in $d$ for all $λ=O_d(1)$.

We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a triangle-free graph and give some consequences of these conjectures in Ramsey theory.

We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)≤R_k(P_n)≤R_k(C_n)≤kn+o(n)$. The upper bound was recently improved by Sárközy who showed that $R_k(C_n)≤\left(k-\frack{16k^3+1}\right)n+o(n)$. Here we show $R_k(C_n) ≤(k-\frac14)n +o(n)$, obtaining the first improvement to the coefficient of the linear term by an absolute constant.

We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of $d$-regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of $K_{d,d}$ maximizes the number of independent sets and the independence polynomial of a $d$-regular graph.

For matchings, this shows that the matching polynomial and the total number of matchings of a $d$-regular graph are maximized by a union of copies of $K_{d,d}$. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markström.

In probabilistic language, our main theorems state that for all $d$-regular graphs and all $λ$, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity $λ$ are maximized by $K_{d,d}$. Our method involves constrained optimization problems over distributions of random variables and applies to all $d$-regular graphs directly, without a reduction to the bipartite case.

We show how to use the recently-developed occupancy method and stability results that follow easily from the method to obtain extremal bounds on the individual coefficients of the partition functions over various classes of bounded degree graphs.

As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström, and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of $q$-colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least $5$.

We develop a theory of regularity inheritance in $3$-uniform hypergraphs. As a simple consequence we deduce a strengthening of a counting lemma of Frankl and Rödl. We believe that the approach is sufficiently flexible and general to permit extensions of our results in the direction of a hypergraph blow-up lemma.

In this thesis we explore extremal graph theory, focusing on new methods which apply to different notions of regular graph. The first notion is $d$-regularity, which means each vertex of a graph is contained in exactly $d$ edges, and the second notion is Szemerédi regularity, which is a strong, approximate version of this property that relates to pseudorandomness.

We begin with a novel method for optimising observables of Gibbs distributions in sparse graphs. The simplest application of the method is to the hard-core model, concerning independent sets in $d$-regular graphs, where we prove a tight upper bound on an observable known as the occupancy fraction. We also cover applications to matchings and colourings, in each case proving a tight bound on an observable of a Gibbs distribution and deriving an extremal result on the number of a relevant combinatorial structure in regular graphs. The results relate to a wide range of topics including statistical physics and Ramsey theory.

We then turn to a form of Szemerédi regularity in sparse hypergraphs, and develop a method for embedding complexes that generalises a widely-applied method for counting in pseudorandom graphs. We prove an inheritance lemma which shows that the neighbourhood of a sparse, regular subgraph of a highly pseudorandom hypergraph typically inherits regularity in a natural way. This shows that we may embed complexes into suitable regular hypergraphs vertex-by-vertex, in much the same way as one can prove a counting lemma for regular graphs.

Finally, we consider the multicolour Ramsey number of paths and even cycles. A well-known density argument shows that when the edges of a complete graph on $kn$ vertices are coloured with $k$ colours, one can find a monochromatic path on $n$ vertices. We give an improvement to this bound by exploiting the structure of the densest colour, and use the regularity method to extend the result to even cycles.