arXiv subjects

Tom Kelly

Publications and source records attributed to Tom Kelly.

At least 37 records · Page 2Linked to original sources

arXiv2023

On the density of critical graphs with no large cliques

A graph $G$ is \textit{$k$-critical} if $χ(G) = k$ and every proper subgraph of $G$ is $(k - 1)$-colorable, and if $L$ is a list-assignment for $G$, then $G$ is \textit{$L$-critical} if $G$ is not $L$-colorable but every proper induced subgraph of $G$ is. In 2014, Kostochka and Yancey proved a lower bound on the average degree of an $n$-vertex $k$-critical graph tending to $k - \frac{2}{k - 1}$ for large $n$ that is tight for infinitely many values of $n$, and they asked how their bound may be improved for graphs not containing a large clique. Answering this question, we prove that for $\varepsilon \leq 2.6\cdot10^{-10}$, if $k$ is sufficiently large and $G$ is a $K_{ω+ 1}$-free $L$-critical graph where $ω\leq k - \log^{10}k$ and $L$ is a list-assignment for $G$ such that $|L(v)| = k - 1$ for all $v\in V(G)$, then the average degree of $G$ is at least $(1 + \varepsilon)(k - 1) - \varepsilon ω- 1$. This result implies that for some $\varepsilon > 0$, for every graph $G$ satisfying $ω(G) \leq \mathrm{mad}(G) - \log^{10}\mathrm{mad}(G)$ where $ω(G)$ is the size of the largest clique in $G$ and $\mathrm{mad}(G)$ is the maximum average degree of $G$, the list-chromatic number of $G$ is at most $\left\lceil (1 - \varepsilon)(\mathrm{mad}(G) + 1) + \varepsilonω(G)\right\rceil$.

math.CO↗

arXiv2023

Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor

We prove that for $n \in \mathbb N$ and an absolute constant $C$, if $p \geq C\log^2 n / n$ and $L_{i,j} \subseteq [n]$ is a random subset of $[n]$ where each $k\in [n]$ is included in $L_{i,j}$ independently with probability $p$ for each $i, j\in [n]$, then asymptotically almost surely there is an order-$n$ Latin square in which the entry in the $i$th row and $j$th column lies in $L_{i,j}$. The problem of determining the threshold probability for the existence of an order-$n$ Latin square was raised independently by Johansson, by Luria and Simkin, and by Casselgren and H{ä}ggkvist; our result provides an upper bound which is tight up to a factor of $\log n$ and strengthens the bound recently obtained by Sah, Sawhney, and Simkin. We also prove analogous results for Steiner triple systems and $1$-factorizations of complete graphs, and moreover, we show that each of these thresholds is at most the threshold for the existence of a $1$-factorization of a nearly complete regular bipartite graph.

math.CO↗

arXiv2023

A proof of the Erdős-Faber-Lovász conjecture

The Erdős-Faber-Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stability versions of this result, which confirm a prediction of Kahn.

math.CO↗

arXiv2022

Large-Scale Auto-Regressive Modeling Of Street Networks

We present a novel generative method for the creation of city-scale road layouts. While the output of recent methods is limited in both size of the covered area and diversity, our framework produces large traversable graphs of high quality consisting of vertices and edges representing complete street networks covering 400 square kilometers or more. While our framework can process general 2D embedded graphs, we focus on street networks due to the wide availability of training data. Our generative framework consists of a transformer decoder that is used in a sliding window manner to predict a field of indices, with each index encoding a representation of the local neighborhood. The semantics of each index is determined by a dictionary of context vectors. The index field is then input to a decoder to compute the street graph. Using data from OpenStreetMap, we train our system on whole cities and even across large countries such as the US, and finally compare it to the state of the art.

cs.LG↗

arXiv2022

Almost all optimally coloured complete graphs contain a rainbow Hamilton path

A subgraph $H$ of an edge-coloured graph is called rainbow if all of the edges of $H$ have different colours. In 1989, Andersen conjectured that every proper edge-colouring of $K_{n}$ admits a rainbow path of length $n-2$. We show that almost all optimal edge-colourings of $K_{n}$ admit both (i) a rainbow Hamilton path and (ii) a rainbow cycle using all of the colours. This result demonstrates that Andersen's Conjecture holds for almost all optimal edge-colourings of $K_{n}$ and answers a recent question of Ferber, Jain, and Sudakov. Our result also has applications to the existence of transversals in random symmetric Latin squares.

math.CO↗

arXiv2022

Hamilton transversals in random Latin squares

Gyárfás and Sárközy conjectured that every $n\times n$ Latin square has a `cycle-free' partial transversal of size $n-2$. We confirm this conjecture in a strong sense for almost all Latin squares, by showing that as $n \rightarrow \infty$, all but a vanishing proportion of $n\times n$ Latin squares have a Hamilton transversal, i.e. a full transversal for which any proper subset is cycle-free. In fact, we prove a counting result that in almost all Latin squares, the number of Hamilton transversals is essentially that of Taranenko's upper bound on the number of full transversals. This result strengthens a result of Kwan (which in turn implies that almost all Latin squares also satisfy the famous Ryser-Brualdi-Stein conjecture).

math.CO↗

arXiv2022

Projective Urban Texturing

This paper proposes a method for automatic generation of textures for 3D city meshes in immersive urban environments. Many recent pipelines capture or synthesize large quantities of city geometry using scanners or procedural modeling pipelines. Such geometry is intricate and realistic, however the generation of photo-realistic textures for such large scenes remains a problem. We propose to generate textures for input target 3D meshes driven by the textural style present in readily available datasets of panoramic photos capturing urban environments. Re-targeting such 2D datasets to 3D geometry is challenging because the underlying shape, size, and layout of the urban structures in the photos do not correspond to the ones in the target meshes. Photos also often have objects (e.g., trees, vehicles) that may not even be present in the target geometry. To address these issues we present a method, called Projective Urban Texturing (PUT), which re-targets textural style from real-world panoramic images to unseen urban meshes. PUT relies on contrastive and adversarial training of a neural architecture designed for unpaired image-to-texture translation. The generated textures are stored in a texture atlas applied to the target 3D mesh geometry. To promote texture consistency, PUT employs an iterative procedure in which texture synthesis is conditioned on previously generated, adjacent textures. We demonstrate both quantitative and qualitative evaluation of the generated textures.

cs.CV↗

arXiv2021

Graph and hypergraph colouring via nibble methods: A survey

This paper provides a survey of methods, results, and open problems on graph and hypergraph colourings, with a particular emphasis on semi-random `nibble' methods. We also give a detailed sketch of some aspects of the recent proof of the Erdős-Faber-Lovász conjecture.

math.CO↗

arXiv2021

Seamless Satellite-image Synthesis

We introduce Seamless Satellite-image Synthesis (SSS), a novel neural architecture to create scale-and-space continuous satellite textures from cartographic data. While 2D map data is cheap and easily synthesized, accurate satellite imagery is expensive and often unavailable or out of date. Our approach generates seamless textures over arbitrarily large spatial extents which are consistent through scale-space. To overcome tile size limitations in image-to-image translation approaches, SSS learns to remove seams between tiled images in a semantically meaningful manner. Scale-space continuity is achieved by a hierarchy of networks conditioned on style and cartographic data. Our qualitative and quantitative evaluations show that our system improves over the state-of-the-art in several key areas. We show applications to texturing procedurally generation maps and interactive satellite image manipulation.

cs.CV↗

arXiv2021

Colourings, transversals and local sparsity

Motivated both by recently introduced forms of list colouring and by earlier work on independent transversals subject to a local sparsity condition, we use the semi-random method to prove the following result. For any function $μ$ satisfying $μ(d)=o(d)$ as $d\to\infty$, there is a function $λ$ satisfying $λ(d)=d+o(d)$ as $d\to\infty$ such that the following holds. For any graph $H$ and any partition of its vertices into parts of size at least $λ$ such that (a) for each part the average over its vertices of degree to other parts is at most $d$, and (b) the maximum degree from a vertex to some other part is at most $μ$, there is guaranteed to be a transversal of the parts that forms an independent set of $H$. This is a common strengthening of two results of Loh and Sudakov (2007) and Molloy and Thron (2012), each of which in turn implies an earlier result of Reed and Sudakov (2002).

math.CO↗

arXiv2021

How Computer Science Can Aid Forest Restoration

The world faces two interlinked crises: climate change and loss of biodiversity. Forest restoration on degraded lands and surplus croplands can play a significant role both in sequestering carbon and re-establishing bio-diversity. There is a considerable body of research and practice that addresses forest restoration. However, there has been little work by computer scientists to bring powerful computational techniques to bear on this important area of work, perhaps due to a lack of awareness. In an attempt to bridge this gap, we present our vision of how techniques from computer science, broadly speaking, can aid current practice in forest restoration.

cs.CY↗

arXiv2021

A local epsilon version of Reed's Conjecture

In 1998, Reed conjectured that every graph $G$ satisfies $χ(G) \leq \lceil \frac{1}{2}(Δ(G) + 1 + ω(G))\rceil$, where $χ(G)$ is the chromatic number of $G$, $Δ(G)$ is the maximum degree of $G$, and $ω(G)$ is the clique number of $G$. As evidence for his conjecture, he proved an "epsilon version" of it, i.e. that there exists some $\varepsilon > 0$ such that $χ(G) \leq (1 - \varepsilon)(Δ(G) + 1) + \varepsilonω(G)$. It is natural to ask if Reed's conjecture or an epsilon version of it is true for the list-chromatic number. In this paper we consider a "local version" of the list-coloring version of Reed's conjecture. Namely, we conjecture that if $G$ is a graph with list-assignment $L$ such that for each vertex $v$ of $G$, $|L(v)| \geq \lceil \frac{1}{2}(d(v) + 1 + ω(v))\rceil$, where $d(v)$ is the degree of $v$ and $ω(v)$ is the size of the largest clique containing $v$, then $G$ is $L$-colorable. Our main result is that an "epsilon version" of this conjecture is true, under some mild assumptions. Using this result, we also prove a significantly improved lower bound on the density of $k$-critical graphs with clique number less than $k/2$, as follows. For every $α> 0$, if $\varepsilon \leq \frac{α^2}{1350}$, then if $G$ is an $L$-critical graph for some $k$-list-assignment $L$ such that $ω(G) < (\frac{1}{2} - α)k$ and $k$ is sufficiently large, then $G$ has average degree at least $(1 + \varepsilon)k$. This implies that for every $α> 0$, there exists $\varepsilon > 0$ such that if $G$ is a graph with $ω(G)\leq (\frac{1}{2} - α)\mathrm{mad}(G)$, where $\mathrm{mad}(G)$ is the maximum average degree of $G$, then $χ_\ell(G) \leq \left\lceil (1 - \varepsilon)(\mathrm{mad}(G) + 1) + \varepsilon ω(G)\right\rceil$.

math.CO↗

arXiv2021

SketchGen: Generating Constrained CAD Sketches

Computer-aided design (CAD) is the most widely used modeling approach for technical design. The typical starting point in these designs is 2D sketches which can later be extruded and combined to obtain complex three-dimensional assemblies. Such sketches are typically composed of parametric primitives, such as points, lines, and circular arcs, augmented with geometric constraints linking the primitives, such as coincidence, parallelism, or orthogonality. Sketches can be represented as graphs, with the primitives as nodes and the constraints as edges. Training a model to automatically generate CAD sketches can enable several novel workflows, but is challenging due to the complexity of the graphs and the heterogeneity of the primitives and constraints. In particular, each type of primitive and constraint may require a record of different size and parameter types. We propose SketchGen as a generative model based on a transformer architecture to address the heterogeneity problem by carefully designing a sequential language for the primitives and constraints that allows distinguishing between different primitive or constraint types and their parameters, while encouraging our model to re-use information across related parameters, encoding shared structure. A particular highlight of our work is the ability to produce primitives linked via constraints that enables the final output to be further regularized via a constraint solver. We evaluate our model by demonstrating constraint prediction for given sets of primitives and full sketch generation from scratch, showing that our approach significantly out performs the state-of-the-art in CAD sketch generation.

cs.LG↗

arXiv2021

Retrofitting Effect Handlers onto OCaml

Effect handlers have been gathering momentum as a mechanism for modular programming with user-defined effects. Effect handlers allow for non-local control flow mechanisms such as generators, async/await, lightweight threads and coroutines to be composably expressed. We present a design and evaluate a full-fledged efficient implementation of effect handlers for OCaml, an industrial-strength multi-paradigm programming language. Our implementation strives to maintain the backwards compatibility and performance profile of existing OCaml code. Retrofitting effect handlers onto OCaml is challenging since OCaml does not currently have any non-local control flow mechanisms other than exceptions. Our implementation of effect handlers for OCaml: (i) imposes a mean 1% overhead on a comprehensive macro benchmark suite that does not use effect handlers; (ii) remains compatible with program analysis tools that inspect the stack; and (iii) is efficient for new code that makes use of effect handlers.

cs.PL↗

arXiv2020

Generative Layout Modeling using Constraint Graphs

We propose a new generative model for layout generation. We generate layouts in three steps. First, we generate the layout elements as nodes in a layout graph. Second, we compute constraints between layout elements as edges in the layout graph. Third, we solve for the final layout using constrained optimization. For the first two steps, we build on recent transformer architectures. The layout optimization implements the constraints efficiently. We show three practical contributions compared to the state of the art: our work requires no user input, produces higher quality layouts, and enables many novel capabilities for conditional layout generation.

cs.CV↗

arXiv2020

Fractional vertex-arboricity of planar graphs

We initiate a systematic study of the fractional vertex-arboricity of planar graphs and demonstrate connections to open problems concerning both fractional coloring and the size of the largest induced forest in planar graphs. In particular, the following three long-standing conjectures concern the size of a largest induced forest in a planar graph, and we conjecture that each of these can be generalized to the setting of fractional vertex-arboricity. In 1979, Albertson and Berman conjectured that every planar graph has an induced forest on at least half of its vertices, in 1987, Akiyama and Watanabe conjectured that every bipartite planar graph has an induced forest on at least five-eighths of its vertices, and in 2010, Kowalik, Lužar, and Škrekovski conjectured that every planar graph of girth at least five has an induced forest on at least seven-tenths of its vertices. We make progress toward the fractional generalization of the latter of these, by proving that every planar graph of girth at least five has fractional vertex-arboricity at most $2 - 1/324$.

math.CO↗

arXiv2020

Retrofitting Parallelism onto OCaml

OCaml is an industrial-strength, multi-paradigm programming language, widely used in industry and academia. OCaml is also one of the few modern managed system programming languages to lack support for shared memory parallel programming. This paper describes the design, a full-fledged implementation and evaluation of a mostly-concurrent garbage collector (GC) for the multicore extension of the OCaml programming language. Given that we propose to add parallelism to a widely used programming language with millions of lines of existing code, we face the challenge of maintaining backwards compatibility--not just in terms of the language features but also the performance of single-threaded code running with the new GC. To this end, the paper presents a series of novel techniques and demonstrates that the new GC strikes a balance between performance and feature backwards compatibility for sequential programs and scales admirably on modern multicore processors.

cs.PL↗

arXiv2019

The structure of binary matroids with no induced claw or Fano plane restriction

An 'induced restriction' of a simple binary matroid $M$ is a restriction $M|F$, where $F$ is a flat of $M$. We consider the class $\mathcal{M}$ of all simple binary matroids $M$ containing neither a free matroid on three elements (which we call a 'claw'), nor a Fano plane as an induced restriction. We give an exact structure theorem for this class; two of its consequences are that the matroids in $\mathcal{M}$ have unbounded critical number, while the matroids in $\mathcal{M}$ not containing the clique $M(K_5)$ as an induced restriction have critical number at most $2$.

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