arXiv subjects

Matthew Kwan

Publications and source records attributed to Matthew Kwan.

63 records · Page 4Linked to original sources

arXiv2018

On Kahn's basis conjecture

In 1991, Kahn made the following conjecture. For any $n$-dimensional vector space $V$ and any $n\times n$ array of $n^2$ bases of $V$, it is possible to choose a representative vector from each of these bases in such a way that the representatives from each row form a basis and the representatives from each column also form a basis. Rota's basis conjecture can be viewed as a special case of Kahn's conjecture, where for each column, all the bases in that column are the same. Recently the authors showed that in the setting of Rota's basis conjecture it is possible to find suitable representatives in $\left(1/2-o\left(1\right)\right)n$ of the rows. In this companion note we give a slight modification of our arguments which generalises this result to the setting of Kahn's conjecture. Our results also apply to the more general setting of matroids.

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arXiv2018

The random k-matching-free process

Let $\mathcal{P}$ be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty $n$-vertex graph and then adds edges one-by-one, each chosen uniformly at random subject to the constraint that $\mathcal{P}$ is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extremal combinatorics, and leading to the discovery of important probabilistic tools. In this paper we consider the $k$-matching-free process, where $\mathcal{P}$ is the property of not containing a matching of size $k$. We are able to analyse the behaviour of this process for a wide range of values of $k$; in particular we prove that if $k=o(n)$ or if $n-2k=o(\sqrt{n}/\log n)$ then this process is likely to terminate in a $k$-matching-free graph with the maximum possible number of edges, as characterised by Erdős and Gallai. We also show that these bounds on $k$ are essentially best possible, and we make a first step towards understanding the behaviour of the process in the intermediate regime.

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arXiv2018

Counting Hamilton cycles in sparse random directed graphs

Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+ω(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically $n!(\log n/n(1+o(1)))^{n}$ directed Hamilton cycles.

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arXiv2017

Resilience for the Littlewood-Offord Problem

Consider the sum $X(ξ)=\sum_{i=1}^n a_iξ_i$, where $a=(a_i)_{i=1}^n$ is a sequence of non-zero reals and $ξ=(ξ_i)_{i=1}^n$ is a sequence of i.i.d. Rademacher random variables (that is, $\Pr[ξ_i=1]=\Pr[ξ_i=-1]=1/2$). The classical Littlewood-Offord problem asks for the best possible upper bound on the concentration probabilities $\Pr[X=x]$. In this paper we study a resilience version of the Littlewood-Offord problem: how many of the $ξ_i$ is an adversary typically allowed to change without being able to force concentration on a particular value? We solve this problem asymptotically, and present a few interesting open problems.

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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.

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arXiv2017

Intercalates and Discrepancy in Random Latin Squares

An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely $N\ge\left(1-o\left(1\right)\right)\,n^{2}/4$, and that $\mathbb{E}N\le\left(1+o\left(1\right)\right)\,n^{2}/2$ (therefore asymptotically almost surely $N\le fn^{2}$ for any $f\to\infty$). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in two fixed rows of a random Latin square. In addition, we discuss a problem of Linial and Luria on low-discrepancy Latin squares.

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arXiv2016

Bounded-degree spanning trees in randomly perturbed graphs

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding trees into fixed dense graphs and into random graphs, and extends a sizeable body of existing research on randomly perturbed graphs. Specifically, we show that there is $c = c(α,Δ)$ such that if G is an n-vertex graph with minimum degree at least $αn$, and T is an n-vertex tree with maximum degree at most $Δ$ , then if we add cn uniformly random edges to G, the resulting graph will contain T asymptotically almost surely (as $n\to\infty$ ). Our proof uses a lemma concerning the decomposition of a dense graph into super-regular pairs of comparable sizes, which may be of independent interest.

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arXiv2016

Cycles and matchings in randomly perturbed digraphs and hypergraphs

We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a dense k-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

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arXiv2014

On the number of spanning trees in random regular graphs

Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

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