arXiv subjects

Richard Montgomery

Publications and source records attributed to Richard Montgomery.

At least 37 records · Page 2Linked to original sources

arXiv2025

Almost every Latin square has a decomposition into transversals

In 1782, Euler conjectured that no Latin square of order $n\equiv 2\; \textrm{mod}\; 4$ has a decomposition into transversals. While confirmed for $n=6$ by Tarry in 1900, Bose, Parker, and Shrikhande constructed counterexamples in 1960 for each $n\equiv 2\; \textrm{mod}\; 4$ with $n\geq 10$. We show that, in fact, counterexamples are extremely common, by showing that if a Latin square of order $n$ is chosen uniformly at random then with high probability it has a decomposition into transversals.

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arXiv2024

Trees with many leaves in tournaments

Sumner's universal tournament conjecture states that every $(2n-2)$-vertex tournament should contain a copy of every $n$-vertex oriented tree. If we know the number of leaves of an oriented tree, or its maximum degree, can we guarantee a copy of the tree with fewer vertices in the tournament? Due to work initiated by Häggkvist and Thomason (for number of leaves) and Kühn, Mycroft and Osthus (for maximum degree), it is known that improvements can be made over Sumner's conjecture in some cases, and indeed sometimes an $(n+o(n))$-vertex tournament may be sufficient. In this paper, we give new results on these problems. Specifically, we show i) for every $α>0$, there exists $n_0\in\mathbb{N}$ such that, whenever $n\geqslant n_0$, every $((1+α)n+k)$-vertex tournament contains a copy of every $n$-vertex oriented tree with $k$ leaves, and ii) for every $α>0$, there exists $c>0$ and $n_0\in\mathbb{N}$ such that, whenever $n\geqslant n_0$, every $(1+α)n$-vertex tournament contains a copy of every $n$-vertex oriented tree with maximum degree $Δ(T)\leqslant cn$. Our first result gives an asymptotic form of a conjecture by Havet and Thomassé, while the second improves a result of Mycroft and Naia which applies to trees with polylogarithmic maximum degree.

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arXiv2024

Transversals in Latin Squares

A Latin square is an $n$ by $n$ grid filled with $n$ symbols so that each symbol appears exactly once in each row and each column. A transversal in a Latin square is a collection of cells which do not share any row, column, or symbol. This survey will focus on results from the last decade which have continued the long history of the study of transversals in Latin squares.

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arXiv2024

The negative energy N-body problem has finite diameter

The Jacobi-Maupertuis metric provides a reformulation of the classical N-body problem as a geodesic flow on an energy-dependent metric space denoted $M_E$ where $E$ is the energy of the problem. We show that $M_E$ has finite diameter for $E < 0$. Consequently $M_E$ has no metric rays. Motivation comes from work of Burgos- Maderna and Polimeni-Terracini for the case $E \ge 0$ and from a need to correct an error made in a previous ``proof''. We show that $M_E$ has finite diameter for $E < 0$ by showing that there is a constant $D$ such that all points of the Hill region lie a distance $D$ from the Hill boundary. (When $E \ge 0$ the Hill boundary is empty.) The proof relies on a game of escape which allows us to quantify the escape rate from a closed subset of configuration space, and the reduction of this game to one of escaping the boundary of a polyhedral convex cone into its interior.

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arXiv2024

Approximate path decompositions of regular graphs

We show that the edges of any $d$-regular graph can be almost decomposed into paths of length roughly $d$, giving an approximate solution to a problem of Kotzig from 1957. Along the way, we show that almost all of the vertices of a $d$-regular graph can be partitioned into $n/(d+1)$ paths, asymptotically confirming a conjecture of Magnant and Martin from 2009.

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arXiv2024

Hamiltonicity of expanders: optimal bounds and applications

An $n$-vertex graph $G$ is a $C$-expander if $|N(X)|\geq C|X|$ for every $X\subseteq V(G)$ with $|X|< n/2C$ and there is an edge between every two disjoint sets of at least $n/2C$ vertices. We show that there is some constant $C>0$ for which every $C$-expander is Hamiltonian. In particular, this implies the well known conjecture of Krivelevich and Sudakov from 2003 on Hamilton cycles in $(n,d,λ)$-graphs. This completes a long line of research on the Hamiltonicity of sparse graphs, and has many applications, including to the Hamiltonicity of random Cayley graphs.

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arXiv2024

Edge-disjoint cycles with the same vertex set

In 1975, Erdős asked for the maximum number of edges that an $n$-vertex graph can have if it does not contain two edge-disjoint cycles on the same vertex set. It is known that Turán-type results can be used to prove an upper bound of $n^{3/2+o(1)}$. However, this approach cannot give an upper bound better than $Ω(n^{3/2})$. We show that, for any $k\geq 2$, every $n$-vertex graph with at least $n \cdot \mathrm{polylog}(n)$ edges contains $k$ pairwise edge-disjoint cycles with the same vertex set, resolving this old problem in a strong form up to a polylogarithmic factor. The well-known construction of Pyber, Rödl and Szemerédi of graphs without $4$-regular subgraphs shows that there are $n$-vertex graphs with $Ω(n\log \log n)$ edges which do not contain two cycles with the same vertex set, so the polylogarithmic term in our result cannot be completely removed. Our proof combines a variety of techniques including sublinear expanders, absorption and a novel tool for regularisation, which is of independent interest. Among other applications, this tool can be used to regularise an expander while still preserving certain key expansion properties.

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arXiv2024

The Honest Embedding Dimension of a Numerical Semigroup

Attached to a singular analytic curve germ in $d$-space is a numerical semigroup: a subset $S$ of the non-negative integers which is closed under addition and whose complement isfinite. Conversely, associated to any numerical semigroup $S$ is a canonical mononial curve in $e$-space where $e$ is the number of minimal generators of the semigroup. It may happen that $d < e = e(S)$ where $S$ is the semigroup of the curve in $d$-space. Define the minimal (or `honest') embedding of a numerical semigroup to be the smallest $d$ such that $S$ is realized by a curve in $d$-space. Problem: characterize the numerical semigroups having minimal embedding dimension $d$. The answer is known for the case $d=2$ of planar curves and reviewed in an Appendix to this paper. The case $d =3$ of the problem is open. Our main result is a characterization of the multiplicity $4$ numerical semigroups whose minimal embedding dimension is $3$. See figure 1. The motivation for this work came from thinking about Legendrian curve singularities.

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arXiv2023

The Kepler Cone, Maclaurin Duality and Jacobi-Maupertuis metrics

The Kepler problem is the special case $α= 1$ of the power law problem: to solve Newton's equations for a central force whose potential is of the form $-μ/r^α$ where $μ$ is a coupling constant. Associated to such a problem is a two-dimensional cone with cone angle $2 πc$ with $c = 1 - \fracα{2}$. We construct a transformation taking the geodesics of this cone to the zero energy solutions of the $α$-power law problem. The `Kepler Cone' is the cone associated to the Kepler problem. This zero-energy cone transformation is a special case of a transformation discovered by Maclaurin in the 1740s transforming the $α$- power law problem for any energies to a `Maclaurin dual' $γ$-power law problem where $γ= \frac{2 α}{2-α}$ and which, in the process, mixes up the energy of one problem with the coupling constant of the other. We derive Maclaurin duality using the Jacobi-Maupertuis metric reformulation of mechanics. We then use the conical metric to explain properties of Rutherford-type scattering off power law potentials at positive energies. The one possibly new result in the paper concerns ``star-burst curves'' which arise as limits of families negative energy solutions as their angular momentum tends to zero. We relate geodesic scattering on the cone to Rutherford type scattering of beams of solutions in the potential. We describe some history around Maclaurin duality and give two derivations of the Jacobi-Maupertuis metric reformulation of classical mechanics. The piece is expository, aimed at an upper-division undergraduate. Think American Math. Monthly.

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arXiv2023

Towards the Erdős-Gallai Cycle Decomposition Conjecture

In the 1960's, Erdős and Gallai conjectured that the edges of any $n$-vertex graph can be decomposed into $O(n)$ cycles and edges. We improve upon the previous best bound of $O(n\log\log n)$ cycles and edges due to Conlon, Fox and Sudakov, by showing an $n$-vertex graph can always be decomposed into $O(n\log^{*}n)$ cycles and edges, where $\log^{*}n$ is the iterated logarithm function.

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arXiv2023

Spanning cycles in random directed graphs

We show that, in almost every $n$-vertex random directed graph process, a copy of every possible $n$-vertex oriented cycle will appear strictly before a directed Hamilton cycle does, except of course for the directed cycle itself. Furthermore, given an arbitrary $n$-vertex oriented cycle, we determine the sharp threshold for its appearance in the binomial random directed graph. These results confirm, in a strong form, a conjecture of Ferber and Long.

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arXiv2023

Ramsey numbers of bounded degree trees versus general graphs

For every $k\ge 2$ and $Δ$, we prove that there exists a constant $C_{Δ,k}$ such that the following holds. For every graph $H$ with $χ(H)=k$ and every tree with at least $C_{Δ,k}|H|$ vertices and maximum degree at most $Δ$, the Ramsey number $R(T,H)$ is $(k-1)(|T|-1)+σ(H)$, where $σ(H)$ is the size of a smallest colour class across all proper $k$-colourings of $H$. This is tight up to the value of $C_{Δ,k}$, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.

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arXiv2023

A proof of the Ryser-Brualdi-Stein conjecture for large even $n$

A Latin square of order $n$ is an $n$ by $n$ grid filled using $n$ symbols so that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of cells which share no symbol, row or column. The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every Latin square of order $n$ contains a transversal with $n-1$ cells, and a transversal with $n$ cells if $n$ is odd. Keevash, Pokrovskiy, Sudakov and Yepremyan recently improved the long-standing best known bounds towards this conjecture by showing that every Latin square of order $n$ has a transversal with $n-O(\log n/\log\log n)$ cells. Here, we show, for sufficiently large $n$, that every Latin square of order $n$ has a transversal with $n-1$ cells. We also apply our methods to show that, for sufficiently large $n$, every Steiner triple system of order $n$ has a matching containing at least $(n-4)/3$ edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and Yepremyan, who found such matchings with $n/3-O(\log n/\log\log n)$ edges, and proves a conjecture of Brouwer from 1981 for large $n$.

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arXiv2023

Counting spanning subgraphs in dense hypergraphs

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell\leq k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least $$\cases{\left(\dfrac{1}{2}+o(1)\right)n & if $(k-\ell)\mid k$,\\ & \\ \left(\dfrac{1}{\lceil \frac{k}{k-\ell}\rceil(k-\ell)}+o(1)\right)n & if $(k-\ell)\nmid k$,}$$ contains $\exp(n\log n-Θ(n))$ Hamilton $\ell$-cycles as long as $(k-\ell)\mid n$. When $(k-\ell)\mid k$ this gives a simple proof of a result of Glock, Gould, Joos, Kühn and Osthus, while, when $(k-\ell)\nmid k$ this gives a weaker count than that given by Ferber, Hardiman and Mond or, when $\ell<k/2$, by Ferber, Krivelevich and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.

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arXiv2023

Compactification of the energy surfaces for n bodies

For n bodies moving in Euclidean d-space under the influence of a homogeneous pair interaction we compactify every center-of-mass energy surface, obtaining a 2d(n -1)-1 - dimensional manifold with corners in the sense of Melrose. After a time change, the flow on this manifold is globally defined and non-trivial on the boundary.

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arXiv2023

No Infinite Spin for Planar Total Collision

The infinite spin problem concerns the rotational behavior of total collision orbits in the $n$-body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent normalized configurations and, in particular, there are circles of normalized central configurations. It's conceivable that by means of an infinite spin, a total collision solution could converge to such a circle instead of to a particular point on it. Here we prove that this is not possible, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, but this is not known in general.) Our proof relies on combining the center manifold theorem with the Lojasiewicz gradient inequality.

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arXiv2022

Geodesics in Jet Space

The space $J^k$ of $k$-jets of a real function of one real variable $x$ admits the structure of Carnot group type. As such, $J^k$ admits a submetry (\sR submersion) onto the Euclidean plane. Horizontal lifts of Euclidean lines (which are the left-translates of horizontal one-parameter subgroups) are thus globally minimizing geodesics on $J^k$. All $J^k$-geodesics, minimizing or not, are constructed from degree $k$ polynomials in $x$ according to Anzaldo-Meneses and Monroy-Peréz, reviewed here. The constant polynomials correspond to the horizontal lifts of lines. Which other polynomials yield globally minimizers and what do these minimizers look like? We give a partial answer. Our methods include constructing an intermediate three-dimensional "magnetic" sub-Riemannian space lying between the jet space and the plane, solving a Hamilton-Jacobi (eikonal) equations on this space, and analyzing period asymptotics associated to period degenerations arising from two-parameter families of these polynomials. Along the way, we conjecture the independence of the cut time of any geodesic on jet space from the starting location on that geodesic.

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arXiv2022

A solution to Erdős and Hajnal's odd cycle problem

In 1981, Erdős and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let $\mathcal{C}(G)$ be the set of cycle lengths in a graph $G$ and let $\mathcal{C}_\text{odd}(G)$ be the set of odd numbers in $\mathcal{C}(G)$. We prove that, if $G$ has chromatic number $k$, then $\sum_{\ell\in \mathcal{C}_\text{odd}(G)}1/\ell\geq (1/2-o_k(1))\log k$. This solves Erdős and Hajnal's odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, Erdős asked whether there is some $d$ such that each graph with chromatic number at least $d$ (or perhaps even only average degree at least $d$) has a cycle whose length is a power of 2. We show that an average degree condition is sufficient for this problem, solving it with methods that apply to a wide range of sequences in addition to the powers of 2. Finally, we use our methods to show that, for every $k$, there is some $d$ so that every graph with average degree at least $d$ has a subdivision of the complete graph $K_k$ in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984.

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