arXiv subjects

Joseph Briggs

Publications and source records attributed to Joseph Briggs.

At least 19 recordsLinked to original sources

arXiv2026

A discrete view of Gromov's filling area conjecture

A compact metric surface $M$ isometrically fills a closed metric curve $C$ if $\partial M=C$ and $d_M(x,y)=d_C(x,y)$ for every $x,y\in C=\partial M$; that is, $M$ does not introduce any ``shortcuts'' between points on its boundary. Gromov's filling area conjecture in differential geometry from 1983 asserts that among all compact, orientable Riemannian surfaces which isometrically fill the Riemannian circle, the one with the smallest surface area is the hemisphere. Gromov demonstrated that this is indeed the case if $M$ is homeomorphic to the disk. While Gromov's conjecture has since been verified in some other cases, the full conjecture remains unresolved. In this paper, we consider a discrete analogue of Gromov's problem, which is likely natural to those who study graph embeddings on arbitrary surfaces. Using standard graph-theoretic tools, such as Menger's theorem, we obtain reasonable asymptotic bounds on this discrete variant. We then demonstrate how these discrete bounds can be translated to the continuous setting, showing that any isometric filling of the Riemannian circle of length $2\pi$ has surface area at least $1.36\pi$ (the hemisphere has surface area $2\pi$). This appears to be the first quantitative lower-bound on Gromov's problem that applies to arbitrary isometric fillings.

math.DG↗

arXiv2025

Degree sequences realizing labelled perfect matchings

Let $n\in \mathbb{N}$ and $d_1 \geq d_2 \geq d_n\geq 1$ be integers. There is characterization of when $(d_1, d_1, \ldots, d_n)$ is the degree sequence of a graph containing a perfect matching, due to results of Lov\'{a}sz (1974) and Erd\H{o}s and Gallai (1960). But \emph{which} perfect matchings can be realized in the labelled graph? Here we find the extremal answers to this question, showing that the sequence $(d_1, d_2, \ldots, d_n)$: (1) can realize a perfect matching iff it can realize $\{(1, n), (2,n-1), \ldots, (n/2, n/2+1)\}$, and; (2) can realize any perfect matching iff it can realize $\{(1, 2), (3,4), \ldots, (n-1, n)\}$. Our main result is a characterization of when (2) occurs, extending the work of Lov\'{a}sz and Erd\H{o}s and Gallai. Separately, we are also able to establish a conjecture of Yin and Busch, Ferrera, Hartke, Jacobsen, Kaul, and West about packing graphic sequences, establishing a degree-sequence analog of the Sauer-Spencer packing theorem. We conjecture an $h$-factor analog of our main result, and discuss implications for packing $h$ disjoint perfect matchings.

math.CO↗

arXiv2024

Facets in the Vietoris--Rips complexes of hypercubes

In this paper, we investigate the facets of the Vietoris--Rips complex $\mathcal{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension $n$. Using Hadamard matrices, we prove that the number of different dimensions of such facets is a super-polynomial function of the scale $r$, assuming that $n$ is sufficiently large. We show also that the $(2r-1)$-th dimensional homology of the complex $\mathcal{VR}(Q_n; r)$ is non-trivial when $n$ is large enough, provided that the Hadamard matrix of order $2r$ exists.

math.AT↗

arXiv2024

Frogs, hats and common subsequences

Write $W^{(n)}$ to mean the $n$-letter word obtained by repeating a fixed word $W$ and let $R_n$ denote a uniformly random $n$-letter word sampled from the same alphabet as $W$. We are interested in the average length of the longest common subsequence between $W^{(n)}$ and $R_n$, which is known to be $\gamma(W)\cdot n+o(n)$ for some constant $\gamma(W)$. Bukh and Cox recently developed an interacting particle system, dubbed the frog dynamics, which can be used to compute the constant $\gamma(W)$ for any fixed word $W$. They successfully analyzed the simplest case of the frog dynamics to find an explicit formula for the constants $\gamma(12\cdots k)$. We continue this study by using the frog dynamics to find an explicit formula for the constants $\gamma(12\cdots kk\cdots 21)$. The frog dynamics in this case is a variation of the PushTASEP on the ring where some clocks are identical. Interestingly, exclusion processes with correlated clocks of this type appear to have not been analyzed before. Our analysis leads to a seemingly new combinatorial object which could be of independent interest: frogs with hats!

math.CO↗

arXiv2024

Phase transitions in isoperimetric problems on the integers

Barber and Erde asked the following question: if $B$ generates $\mathbb Z^d$ as an additive group, then must the extremal sets for the vertex/edge-isoperimetric inequality on the Cayley graph $\operatorname{Cay}(\mathbb Z^d,B)$ form a nested family? We answer this question negatively for both the vertex- and edge-isoperimetric inequalities, specifically in the case of $d=1$. The key is to show that the structure of the cylinder $\mathbb Z\times(\mathbb Z/k\mathbb Z)$ can be mimicked in certain Cayley graphs on $\mathbb Z$, leading to a phase transition. We do, however, show that Barber--Erde's question for Cayley graphs on $\mathbb Z$ has a positive answer if one is allowed to ignore finitely many sets.

math.CO↗

arXiv2023

Looms

A pair $(A,B)$ of hypergraphs is called orthogonal if $|a \cap b|=1$ for every pair of edges $a \in A$ and $b \in B$. An orthogonal pair of hypergraphs is called a loom if each of its two members is the set of minimum covers of the other. Looms appear naturally in the context of a conjecture of Gy\'arf\'as and Lehel on the covering number of cross-intersecting hypergraphs. We study their properties and ways of construction, and prove special cases of a conjecture that if true would imply the Gy\'arf\'as--Lehel conjecture.

math.CO↗

arXiv2023

Transversals and colorings of simplicial spheres

Motivated from the surrounding property of a point set in $\mathbb{R}^d$ introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the maximum transversal ratio of simplicial $d$-spheres, we provide two infinite constructions. The first construction gives infintely many $(d+1)$-dimensional simplicial polytopes with the transversal ratio exactly $\frac{2}{d+2}$ for every $d\geq 2$. In the case of $d=2$, this meets the previously well-known upper bound $1/2$ tightly. The second gives infinitely many simplicial 3-spheres with the transversal ratio greater than $1/2$. This was unexpected from what was previously known about the surrounding property. Moreover, we show that, for $d\geq 3$, the facet hypergraph $\mathcal{F}(\mathsf{K})$ of a $d$-dimensional simplicial sphere $\mathsf{K}$ has the chromatic number $χ(\mathcal{F}(\mathsf{K})) \in O(n^{\frac{\lceil d/2\rceil-1}{d}})$, where $n$ is the number of vertices of $\mathsf{K}$. This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz.

math.CO↗

arXiv2022

A Proof of the $(n,k,t)$-Conjectures

An \emph{$(n,k,t)$-graph} is a graph on $n$ vertices in which every set of $k$ vertices contains a clique on $t$ vertices. Tur\'an's Theorem, rephrased in terms of graph complements, states that the unique minimum $(n,k,2)$-graph is an equitable disjoint union of cliques. We prove that minimum $(n,k,t)$-graphs are always disjoint unions of cliques for any $t$ (despite \allowbreak nonuniqueness of extremal examples), thereby generalizing Tur\'an's Theorem and confirming two conjectures of Hoffman et al.

math.CO↗

arXiv2022

Fractionally balanced hypergraphs and rainbow KKM theorems

A d-partite hypergraph is called *fractionally balanced* if there exists a non-negative, not identically zero, function on its edge set that has constant degrees in each vertex side. Using a topological version of Hall's theorem we prove lower bounds on the matching number of such hypergraphs. These bounds yield rainbow versions of the KKM theorem for products of simplices, which in turn are used to obtain some results on multiple-cake division, and on rainbow matchings in families of d-intervals.

math.CO↗

arXiv2021

Cooperative conditions for the existence of rainbow matchings

Let $k>1$, and let $\mathcal{F}$ be a family of $2n+k-3$ non-empty sets of edges in a bipartite graph. If the union of every $k$ members of $\mathcal{F}$ contains a matching of size $n$, then there exists an $\mathcal{F}$-rainbow matching of size $n$. Replacing $2n+k-3$ by $2n+k-2$, the result is true also for $k=1$, and it can be proved (for all $k$) both topologically and by a relatively simple combinatorial argument. The main effort is in gaining the last $1$, which makes the result sharp.

math.CO↗

arXiv2021

Rainbow odd cycles

We prove that every family of (not necessarily distinct) odd cycles $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$'s, forming an odd cycle). As part of the proof, we characterize those families of $n$ odd cycles in $K_{n+1}$ that do not have any rainbow odd cycle. We also characterize those families of $n$ cycles in $K_{n+1}$, as well as those of $n$ edge-disjoint nonempty subgraphs of $K_{n+1}$, without any rainbow cycle.

math.CO↗

arXiv2021

Choice Functions

This is a survey paper on rainbow sets (another name for ``choice functions''). The main theme is the distinction between two types of choice functions: those having a large (in the sense of belonging to some specified filter, namely closed up set of sets) image, and those that have a large domain and small image, where ``smallness'' means belonging to some specified complex (a closed-down set). The paper contains some new results: (1) theorems on scrambled versions, in which the sets are re-shuffled before choosing the rainbow set, and (2) results on weighted and cooperative versions - to be defined below.

math.CO↗

arXiv2021

Badges and rainbow matchings

Drisko proved that $2n-1$ matchings of size $n$ in a bipartite graph have a rainbow matching of size $n$. For general graphs it is conjectured that $2n$ matchings suffice for this purpose (and that $2n-1$ matchings suffice when $n$ is even). The known graphs showing sharpness of this conjecture for $n$ even are called badges. We improve the previously best known bound from $3n-2$ to $3n-3$, using a new line of proof that involves analysis of the appearance of badges. We also prove a "cooperative" generalization: for $t>0$ and $n \geq 3$, any $3n-4+t$ sets of edges, the union of every $t$ of which contains a matching of size $n$, have a rainbow matching of size $n$.

math.CO↗

arXiv2020

Restricted online Ramsey numbers of matchings and trees

Consider a two-player game between players Builder and Painter. Painter begins the game by picking a coloring of the edges of $K_n$, which is hidden from Builder. In each round, Builder points to an edge and Painter reveals its color. Builder's goal is to locate a particular monochromatic structure in Painter's coloring by revealing the color of as few edges as possible. The fewest number of turns required for Builder to win this game is known as the restricted online Ramsey number. In this paper, we consider the situation where this "particular monochromatic structure" is a large matching or a large tree. We show that in any $t$-coloring of $E(K_n)$, Builder can locate a monochromatic matching on at least ${n-t+1\over t+1}$ edges by revealing at most $O(n\log t)$ edges. We show also that in any $3$-coloring of $E(K_n)$, Builder can locate a monochromatic tree on at least $n/2$ vertices by revealing at most $5n$ edges.

math.CO↗

arXiv2020

Particle swarming of sensor correction filters

Reducing the impact of seismic activity on the motion of suspended optics is essential for the operation of ground-based gravitational wave detectors. During periods of increased seismic activity, low-frequency ground translation and tilt cause the Advanced LIGO observatories to lose `lock', reducing their duty cycles. This paper applies modern global-optimisation algorithms to aid in the design of the `sensor correction' filter, used in the control of the active platforms. It is shown that a particle swarm algorithm that minimises a cost-function approximating the differential RMS velocity between platforms can produce control filters that perform better across most frequencies in the control bandwidth than those currently installed. These tests were conducted using training data from the LIGO Hanford Observatory seismic instruments and simulations of the HAM-ISI (Horizontal Access Module Internal Seismic Isolation) platforms. These results show that new methods of producing control filters are ready for use at LIGO. The filters were implemented at LIGO's Hanford Observatory, and use the resulting data to refine the cost function.

astro-ph.IM↗

arXiv2020

A Cryogenic Silicon Interferometer for Gravitational-wave Detection

The detection of gravitational waves from compact binary mergers by LIGO has opened the era of gravitational wave astronomy, revealing a previously hidden side of the cosmos. To maximize the reach of the existing LIGO observatory facilities, we have designed a new instrument that will have 5 times the range of Advanced LIGO, or greater than 100 times the event rate. Observations with this new instrument will make possible dramatic steps toward understanding the physics of the nearby universe, as well as observing the universe out to cosmological distances by the detection of binary black hole coalescences. This article presents the instrument design and a quantitative analysis of the anticipated noise floor.

astro-ph.IM↗

arXiv2019

Rainbow independent sets in certain classes of graphs

For a given class $\mathcal{C}$ of graphs and given integers $m \leq n$, let $f_\mathcal{C}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in any graph belonging to $\mathcal{C}$ have a (possibly partial) rainbow independent $m$-set. Motivated by known results on the finiteness and actual value of $f_\mathcal{C}(n,m)$ when $\mathcal{C}$ is the class of line graphs of graphs, we study this function for various other classes.

math.CO↗

arXiv2019

Choice functions in the intersection of matroids

We prove a common generalization of two results, one on rainbow fractional matchings and one on rainbow sets in the intersection of two matroids: Given $d = r \lceil k \rceil - r + 1$ functions of size (=sum of values) $k$ that are all independent in each of $r$ given matroids, there exists a rainbow set of $supp(f_i)$, $i \leq d$, supporting a function with the same properties.

math.CO↗