arXiv subjects

Catherine Greenhill

Publications and source records attributed to Catherine Greenhill.

At least 37 records · Page 2Linked to original sources

arXiv2020

Subgraph counts for dense random graphs with specified degrees

We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence $(d_1,\ldots,d_n)$ as $n \rightarrow \infty$. We also determine the expected number of spanning trees in this model. The range of degrees covered includes $d_j = λn + O(n^{1/2+\varepsilon})$ for some $λ$ bounded away from $0$ and $1$.

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arXiv2020

The average number of spanning hypertrees in sparse uniform hypergraphs

An $r$-uniform hypergraph $H$ consists of a set of vertices $V$ and a set of edges whose elements are $r$-subsets of $V$. We define a hypertree to be a connected hypergraph which contains no cycles. A hypertree spans a hypergraph $H$ if it is a subhypergraph of $H$ which contains all vertices of $H$. Greenhill, Isaev, Kwan and McKay (2017) gave an asymptotic formula for the average number of spanning trees in graphs with given, sparse degree sequence. We prove an analogous result for $r$-uniform hypergraphs with given degree sequence $\boldsymbol{k} = (k_1,\ldots, k_n)$. Our formula holds when $r^5 k_{\max}^3 = o((kr-k-r)n)$, where $k$ is the average degree and $k_{\max}$ is the maximum degree.

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arXiv2020

Threshold functions for substructures in random subsets of finite vector spaces

The study of substructures in random objects has a long history, beginning with Erdős and Rényi's work on subgraphs of random graphs. We study the existence of certain substructures in random subsets of vector spaces over finite fields. First we provide a general framework which can be applied to establish coarse threshold results and prove a limiting Poisson distribution at the threshold scale. To illustrate our framework we apply our results to $k$-term arithmetic progressions, sums, right triangles, parallelograms and affine planes. We also find coarse thresholds for the property that a random subset of a finite vector space is sum-free, or is a Sidon set.

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arXiv2019

A threshold result for loose Hamiltonicity in random regular uniform hypergraphs

Let $\mathcal{G}(n,r,s)$ denote a uniformly random $r$-regular $s$-uniform hypergraph on $n$ vertices, where $s$ is a fixed constant and $r=r(n)$ may grow with $n$. An $\ell$-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely $\ell$ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles. When $r,s\geq 3$ are fixed integers, we establish a threshold result for the property of containing a loose Hamilton cycle. This partially verifies a conjecture of Dudek, Frieze, Rucinski and Sileikis (2015). In this setting, we also find the asymptotic distribution of the number of loose Hamilton cycles in $\mathcal{G}(n,r,s)$. Finally we prove that for $\ell = 2,\ldots, s-1$ and for $r$ growing moderately as $n\to\infty$, the probability that $\mathcal{G}(n,r,s)$ has a $\ell$-overlapping Hamilton cycle tends to zero.

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arXiv2019

Uniform generation of spanning regular subgraphs of a dense graph

Let $H_n$ be a graph on $n$ vertices and let $\ber{H_n}$ denote the complement of $H_n$. Suppose that $Δ= Δ(n)$ is the maximum degree of $\ber{H_n}$. We analyse three algorithms for sampling $d$-regular subgraphs ($d$-factors) of $H_n$. This is equivalent to uniformly sampling $d$-regular graphs which avoid a set $E(\ber{H_n})$ of forbidden edges. Here $d=d(n)$ is a positive integer which may depend on $n$. Two of these algorithms produce a uniformly random $d$-factor of $H_n$ in expected runtime which is linear in $n$ and low-degree polynomial in $d$ and $Δ$. The first algorithm applies when $(d+Δ)dΔ= o(n)$. This improves on an earlier algorithm by the first author, which required constant $d$ and at most a linear number of edges in $\ber{H_n}$. The second algorithm applies when $H_n$ is regular and $d^2+Δ^2 = o(n)$, adapting an approach developed by the first author together with Wormald. The third algorithm is a simplification of the second, and produces an approximately uniform $d$-factor of $H_n$ in time $O(dn)$. Here the output distribution differs from uniform by $o(1)$ in total variation distance, provided that $d^2+Δ^2 = o(n)$.

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arXiv2019

Rigid colourings of hypergraphs and contiguity

We consider the problem of $q$-colouring a $k$-uniform random hypergraph, where $q,k \geq 3$, and determine the rigidity threshold. For edge densities above the rigidity threshold, we show that almost all solutions have a linear number of vertices that are linearly frozen, meaning that they cannot be recoloured by a sequence of colourings that each change the colour of a sublinear number of vertices. When the edge density is below the threshold, we prove that all but a vanishing proportion of the vertices can be recoloured by a sequence of colourings that recolour only one vertex at a time. This change in the geometry of the solution space has been hypothesised to be the cause of the algorithmic barrier faced by naive colouring algorithms. Our calculations verify predictions made by statistical physicists using the non-rigorous cavity method. The traditional model for problems of this type is the random colouring model, where a random hypergraph is chosen and then a random colouring of that hypergraph is selected. However, it is often easier to work with the planted model, where a random colouring is selected first, and then edges are randomly chosen which respect the colouring. As part of our analysis, we show that up to the condensation phase transition, the random colouring model is contiguous with respect to the planted model. This result is of independent interest.

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arXiv2019

Triangle-creation processes on cubic graphs

An edge switch is an operation which makes a local change in a graph while maintaining the degree of every vertex. We introduce a switch move, called a triangle switch, which creates or deletes at least one triangle. Specifically, a make move is a triangle switch which chooses a path $zwvxy$ of length 4 and replaces it by a triangle $vxwv$ and an edge $yz$, while a break move performs the reverse operation. We consider various Markov chains which perform random triangle switches, and assume that every possible make or break move has positive probability of being performed. Our first result is that any such Markov chain is irreducible on the set of all 3-regular graphs with vertex set $\{1,2,\ldots, n\}$. For a particular, natural Markov chain of this type, we obtain a non-trivial linear upper and lower bounds on the number of triangles in the long run. These bounds are almost surely obtained in linear time, irrespective of the starting graph.

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arXiv2018

Enumerating sparse uniform hypergraphs with given degree sequence and forbidden edges

For $n\geq 3$ and $r=r(n) \geq 3$, let $\boldsymbol{k} =\boldsymbol{k}(n)=(k_1, \ldots, k_n)$ be a sequence of non-negative integers with sum $M(\boldsymbol{k})=\sum_{j=1}^{n} k_j$. We assume that $M(\boldsymbol{k})$ is divisible by $r$ for infinitely many values of $n$, and restrict our attention to these values. Let $X=X(n)$ be a simple $r$-uniform hypergraph on the vertex set $V=\{v_1,v_2, \ldots, v_n\}$ with $t$ edges and maximum degree $x_{\max}$. We denote by $\mathcal{H}_r(\boldsymbol{k})$ the set of all simple $r$-uniform hypergraphs on the vertex set $V$ with degree sequence $\boldsymbol{k}$, and let $\mathcal{H}_r(\boldsymbol{k},X)$ be the set of all hypergraphs in $\mathcal{H}_r(\boldsymbol{k})$ which contain no edge of $X$. We give an asymptotic enumeration formula for the size of $\mathcal{H}_r(\boldsymbol{k},X)$. This formula holds when $r^4 k_{\max}^3=o(M(\boldsymbol{k}))$, $t\, k_{\max}^{3}\, =o(M(\boldsymbol{k})^2)$ and $r\,t\,k_{\max}^4 = o(M(\boldsymbol{k})^3)$. Our proof involves the switching method. As a corollary, we obtain an asymptotic formula for the number of hypergraphs in $\mathcal{H}_r(\boldsymbol{k})$ which contain every edge of $X$. We apply this result to find asymptotic expressions for the expected number of perfect matchings and loose Hamilton cycles in a random hypergraph in $\mathcal{H}_r(\boldsymbol{k})$ in the regular case.

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arXiv2018

The flip Markov chain for connected regular graphs

Mahlmann and Schindelhauer (2005) defined a Markov chain which they called $k$-Flipper, and showed that it is irreducible on the set of all connected regular graphs of a given degree (at least 3). We study the 1-Flipper chain, which we call the flip chain, and prove that the flip chain converges rapidly to the uniform distribution over connected $2r$-regular graphs with $n$ vertices, where $n\geq 8$ and $r = r(n)\geq 2$. Formally, we prove that the distribution of the flip chain will be within $\varepsilon$ of uniform in total variation distance after $\text{poly}(n,r,\log(\varepsilon^{-1}))$ steps. This polynomial upper bound on the mixing time is given explicitly, and improves markedly on a previous bound given by Feder et al.(2006). We achieve this improvement by using a direct two-stage canonical path construction, which we define in a general setting. This work has applications to decentralised networks based on random regular connected graphs of even degree, as a self-stabilising protocol in which nodes spontaneously perform random flips in order to repair the network.

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arXiv2018

Hypergraph coloring up to condensation

Improving a result of Dyer, Frieze and Greenhill [Journal of Combinatorial Theory, Series B, 2015], we determine the $q$-colorability threshold in random $k$-uniform hypergraphs up to an additive error of $\ln 2+\varepsilon_q$, where $\lim_{q\to\infty}\varepsilon_q=0$. The new lower bound on the threshold matches the "condensation phase transition" predicted by statistical physics considerations [Krzakala et al., PNAS 2007].

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arXiv2017

The switch Markov chain for sampling irregular graphs and digraphs

The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences, both in the undirected and directed setting. We prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree $d_{\max}$ which satisfies $3\leq d_{\max}\leq \frac{1}{3}\, \sqrt{M}$, where $M$ is the sum of the degrees. The mixing time bound obtained is only a factor $n$ larger than that established in the regular case, where $n$ is the number of vertices. Our result covers a wide range of degree sequences, including power-law graphs with parameter $γ> 5/2$ and sufficiently many edges. For directed degree sequences such that the switch chain is irreducible, we prove that the switch chain is rapidly mixing when all in-degrees and out-degrees are positive and bounded above by $\frac{1}{4}\, \sqrt{m}$, where $m$ is the number of arcs, and not all in-degrees and out-degrees equal 1. The mixing time bound obtained in the directed case is an order of $m^2$ larger than that established in the regular case.

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arXiv2017

The average number of spanning trees in sparse graphs with given degrees

We give an asymptotic expression for the expected number of spanning trees in a random graph with a given degree sequence $\boldsymbol{d}=(d_1,\ldots, d_n)$, provided that the number of edges is at least $n + \textstyle{\frac{1}{2}} d_{\max}^4$, where $d_{\max}$ is the maximum degree. A key part of our argument involves establishing a concentration result for a certain family of functions over random trees with given degrees, using Prüfer codes.

math.CO↗

arXiv2016

Asymptotic enumeration of sparse uniform linear hypergraphs with given degrees

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main result is an asymptotic enumeration formula for linear $r$-uniform hypergraphs with degree sequence $\boldsymbol{k}$. This formula holds whenever the maximum degree $k_{\max}$ satisfies $r^4 k_{\max}^4(k_{\max} + r) = o(M)$. Our approach is to work with the incidence matrix of a hypergraph, interpreted as the biadjacency matrix of a bipartite graph, enabling us to apply known enumeration results for bipartite graphs. This approach also leads to a new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees, and a result regarding the girth of random bipartite graphs with specified degrees.

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arXiv2015

Asymptotic enumeration of sparse uniform hypergraphs with given degrees

Let $r \geq 2$ be a fixed integer. For infinitely many $n$, let $\boldsymbol{k} = (k_1,..., k_n)$ be a vector of nonnegative integers such that their sum $M$ is divisible by $r$. We present an asymptotic enumeration formula for simple $r$-uniform hypergraphs with degree sequence $k$. (Here "simple" means that all edges are distinct and no edge contains a repeated vertex.) Our formula holds whenever the maximum degree $k_{\mathrm{max}}$ satisfies $k_{\mathrm{max}}^3 = o(M)$.

math.CO↗

arXiv2015

On the chromatic number of a random hypergraph

We consider the problem of $k$-colouring a random $r$-uniform hypergraph with $n$ vertices and $cn$ edges, where $k$, $r$, $c$ remain constant as $n$ tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case $r=2$, must have one of two easily computable values as $n$ tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs.

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arXiv2014

The switch Markov chain for sampling irregular graphs

The problem of efficiently sampling from a set of(undirected) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences. We prove that the switch chain is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree $d_{\max}$ which satisfies $3\leq d_{\max}\leq \frac{1}{4}\, \sqrt{M}$, where $M$ is the sum of the degrees. The mixing time bound obtained is only an order of $n$ larger than that established in the regular case, where $n$ is the number of vertices.

cs.DS↗

arXiv2014

Mixing of the Glauber dynamics for the ferromagnetic Potts model

We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree ($Δ$) of the underlying graph and the number of colours or spins ($q$) in determining whether the dynamics mixes rapidly or not. We find a lower bound $L$ on the number of colours such that Glauber dynamics is rapidly mixing if at least $L$ colours are used. We give a closely-matching upper bound $U$ on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random $Δ$-regular graphs when at most $U$ colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree $Δ$, e.g. toroidal grids for $Δ= 4$.

cs.DM↗

arXiv2014

Structure and eigenvalues of heat-bath Markov chains

We prove that heat-bath chains (which we define in a general setting) have no negative eigenvalues. Two applications of this result are presented: one to single-site heat-bath chains for spin systems and one to a heat-bath Markov chain for sampling contingency tables. Some implications of our main result for the analysis of the mixing time of heat-bath Markov chains are discussed. We also prove an alternative characterisation of heat-bath chains, and consider possible generalisations.

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