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Jonathan A. Noel

Publications and source records attributed to Jonathan A. Noel.

At least 37 records · Page 2Linked to original sources
arXiv2023

Off-Diagonal Commonality of Graphs via Entropy

A graph $H$ is common if the limit as $n\to\infty$ of the minimum density of monochromatic labelled copies of $H$ in an edge colouring of $K_n$ with red and blue is attained by a sequence of quasirandom colourings. We apply an information-theoretic approach to show that certain graphs obtained from odd cycles and paths via gluing operations are common. In fact, for every pair $(H_1,H_2)$ of such graphs, there exists $p\in(0,1)$ such that an appropriate linear combination of red copies of $H_1$ and blue copies of $H_2$ is minimized by a quasirandom colouring in which $p\binom{n}{2}$ edges are red; such a pair $(H_1,H_2)$ is said to be $(p,1-p)$-common. Our approach exploits a strengthening of the common graph property for odd cycles that was recently proved using Schur convexity. We also exhibit a $(p,1-p)$-common pair $(H_1,H_2)$ such that $H_2$ is uncommon.

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arXiv2023

On the Running Time of Hypergraph Bootstrap Percolation

Given $r\geq2$ and an $r$-uniform hypergraph $F$, the $F$-bootstrap process starts with an $r$-uniform hypergraph $H$ and, in each time step, every hyperedge which "completes" a copy of $F$ is added to $H$. The maximum running time of this process has been recently studied in the case that $r=2$ and $F$ is a complete graph by Bollobás, Przykucki, Riordan and Sahasrabudhe [Electron. J. Combin. 24(2) (2017), Paper No. 2.16], Matzke [arXiv:1510.06156v2] and Balogh, Kronenberg, Pokrovskiy and Szabó [arXiv:1907.04559v1]. We consider the case that $r\geq3$ and $F$ is the complete $r$-uniform hypergraph on $k$ vertices. Our main results are that the maximum running time is $Θ\left(n^r\right)$ if $k\geq r+2$ and $Ω\left(n^{r-1}\right)$ if $k=r+1$. For the case $k=r+1$, we conjecture that our lower bound is optimal up to a constant factor when $r=3$, but suspect that it can be improved by more than a constant factor for large $r$.

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arXiv2023

Common Pairs of Graphs

A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a red/blue edge colouring of a large complete graph is asymptotically minimized by a random colouring with an equal proportion of each colour. We extend this notion to an asymmetric setting. That is, we define a pair $(H_1,H_2)$ of graphs to be $(p,1-p)$-common if a particular linear combination of the density of $H_1$ in red and $H_2$ in blue is asymptotically minimized by a random colouring in which each edge is coloured red with probability $p$ and blue with probability $1-p$. We extend many of the results on common graphs to this asymmetric setting. In addition, we obtain several novel results for common pairs of graphs with no natural analogue in the symmetric setting. We also obtain new examples of common graphs in the classical sense and propose several open problems.

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arXiv2023

Extremal Bounds for Three-Neighbour Bootstrap Percolation in Dimensions Two and Three

For $r\geq1$, the $r$-neighbour bootstrap process in a graph $G$ starts with a set of infected vertices and, in each time step, every vertex with at least $r$ infected neighbours becomes infected. The initial infection percolates if every vertex of $G$ is eventually infected. We exactly determine the minimum cardinality of a set that percolates for the $3$-neighbour bootstrap process when $G$ is a $3$-dimensional grid with minimum side-length at least $11$. We also characterize the integers $a$ and $b$ for which there is a set of cardinality $\frac{ab+a+b}{3}$ that percolates for the $3$-neighbour bootstrap process in the $a\times b$ grid; this solves a problem raised by Benevides, Bermond, Lesfari and Nisse [HAL Research Report 03161419v4, 2021].

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arXiv2022

Density maximizers of layered permutations

A permutation is layered if it contains neither 231 nor 312 as a pattern. It is known that, if $σ$ is a layered permutation, then the density of $σ$ in a permutation of order $n$ is maximized by a layered permutation. Albert, Atkinson, Handley, Holton and Stromquist [Electron. J. Combin. 9 (2002), R#5] claimed that the density of a layered permutation with layers of sizes $(a,1,b)$ where $a,b\geq2$ is asymptotically maximized by layered permutations with a bounded number of layers, and conjectured that the same holds if a layered permutation has no consecutive layers of size one and its first and last layers are of size at least two. We show that, if $σ$ is a layered permutation whose first layer is sufficiently large and second layer is of size one, then the number of layers tends to infinity in every sequence of layered permutations asymptotically maximizing the density of $σ$. This disproves the conjecture and the claim of Albert et al. We complement this result by giving sufficient conditions on a layered permutation to have asymptotic or exact maximizers with a bounded number of layers.

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arXiv2022

Characterization of quasirandom permutations by a pattern sum

It is known that a sequence Pi_i of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Pi_i converges to 1/24. We show that there is a set S of 4-point permutations such that the sum of the pattern densities of the permutations from S in the permutations Pi_i converges to |S|/24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight.

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arXiv2021

Pirates in Wonderland: Liquid Democracy has Bicriteria Guarantees

Liquid democracy has a natural graphical representation, the delegation graph. Consequently, the strategic aspects of liquid democracy can be studied as a game over delegation graphs, called the liquid democracy game. Our main result is that this game has bicriteria approximation guarantees, in terms of both rationality and social welfare. Specifically, we prove the price of stability for $ε$-Nash equilibria is exactly $ε$ in the liquid democracy game.

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arXiv2021

Recolouring Homomorphisms to triangle-free reflexive graphs

For a graph $H$, the $H$-recolouring problem $\operatorname{Recol}(H)$ asks, for two given homomorphisms from a given graph $G$ to $H$, if one can get between them by a sequence of homomorphisms of $G$ to $H$ in which consecutive homomorphisms differ on only one vertex. We show that, if $G$ and $H$ are reflexive and $H$ is triangle-free, then this problem can be solved in polynomial time. This shows, at the same time, that the closely related $H$-reconfiguration problem $\operatorname{Recon}(H)$ of deciding whether two given homomorphisms from a given graph $G$ to $H$ are in the same component of the Hom-graph $\operatorname{Hom}(G,H)$, can be solved in polynomial time for triangle-free reflexive graphs $H$.

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arXiv2020

A Sharp Threshold for Bootstrap Percolation in a Random Hypergraph

Given a hypergraph $\mathcal{H}$, the $\mathcal{H}$-bootstrap process starts with an initial set of infected vertices of $\mathcal{H}$ and, at each step, a healthy vertex $v$ becomes infected if there exists a hyperedge of $\mathcal{H}$ in which $v$ is the unique healthy vertex. We say that the set of initially infected vertices percolates if every vertex of $\mathcal{H}$ is eventually infected. We show that this process exhibits a sharp threshold when $\mathcal{H}$ is a hypergraph obtained by randomly sampling hyperedges from an approximately $d$-regular $r$-uniform hypergraph satisfying some mild degree and codegree conditions; this confirms a conjecture of Morris. As a corollary, we obtain a sharp threshold for a variant of the graph bootstrap process for strictly $2$-balanced graphs which generalises a result of Korándi, Peled and Sudakov. Our approach involves an application of the differential equations method.

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arXiv2020

Finitely forcible graphons with an almost arbitrary structure

Graphons are analytic objects representing convergent sequences of large graphs. A graphon is said to be finitely forcible if it is determined by finitely many subgraph densities, i.e., if the asymptotic structure of graphs represented by such a graphon depends only on finitely many density constraints. Such graphons appear in various scenarios, particularly in extremal combinatorics. Lovasz and Szegedy conjectured that all finitely forcible graphons possess a simple structure. This was disproved in a strong sense by Cooper, Kral and Martins, who showed that any graphon is a subgraphon of a finitely forcible graphon. We strenghten this result by showing for every $\varepsilon>0$ that any graphon spans a $1-\varepsilon$ proportion of a finitely forcible graphon.

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arXiv2019

Reconfiguring Graph Homomorphisms on the Sphere

Given a loop-free graph $H$, the reconfiguration problem for homomorphisms to $H$ (also called $H$-colourings) asks: given two $H$-colourings $f$ of $g$ of a graph $G$, is it possible to transform $f$ into $g$ by a sequence of single-vertex colour changes such that every intermediate mapping is an $H$-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs $H$ (e.g. all $C_4$-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever $H$ is a $K_{2,3}$-free quadrangulation of the $2$-sphere (equivalently, the plane) which is not a $4$-cycle. From this result, we deduce an analogous statement for non-bipartite $K_{2,3}$-free quadrangulations of the projective plane. This include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and $4$-chromatic generalized Mycielski graphs, for which it was not. If we instead consider graphs $G$ and $H$ with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for $H$-colourings is PSPACE-complete whenever $H$ is a reflexive $K_{4}$-free triangulation of the $2$-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which $H$-Recolouring is known to be PSPACE-complete for reflexive instances.

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arXiv2019

Cycles of length three and four in tournaments

Linial and Morgenstern conjectured that, among all $n$-vertex tournaments with $d\binom{n}{3}$ cycles of length three, the number of cycles of length four is asymptotically minimized by a random blow-up of a transitive tournament with all but one part of equal size and one smaller part. We prove the conjecture for $d\ge 1/36$ by analyzing the possible spectrum of adjacency matrices of tournaments. We also demonstrate that the family of extremal examples is broader than expected and give its full description for $d\ge 1/16$.

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arXiv2017

Graph Homomorphism Reconfiguration and Frozen $H$-Colourings

For a fixed graph $H$, the reconfiguration problem for $H$-colourings (i.e. homomorphisms to $H$) asks: given a graph $G$ and two $H$-colourings $φ$ and $ψ$ of $G$, does there exist a sequence $f_0,\dots,f_m$ of $H$-colourings such that $f_0=φ$, $f_m=ψ$ and $f_i(u)f_{i+1}(v)\in E(H)$ for every $0\leq i<m$ and $uv\in E(G)$? If the graph $G$ is loop-free, then this is the equivalent to asking whether it possible to transform $φ$ into $ψ$ by changing the colour of one vertex at a time such that all intermediate mappings are $H$-colourings. In the affirmative, we say that $φ$ reconfigures to $ψ$. Currently, the complexity of deciding whether an $H$-colouring $φ$ reconfigures to an $H$-colouring $ψ$ is only known when $H$ is a clique, a circular clique, a $C_4$-free graph, or in a few other cases which are easily derived from these. We show that this problem is PSPACE-complete when $H$ is an odd wheel. An important notion in the study of reconfiguration problems for $H$-colourings is that of a frozen $H$-colouring; i.e. an $H$-colouring $φ$ such that $φ$ does not reconfigure to any $H$-colouring $ψ$ such that $ψ\neq φ$. We obtain an explicit dichotomy theorem for the problem of deciding whether a given graph $G$ admits a frozen $H$-colouring. The hardness proof involves a reduction from a CSP problem which is shown to be NP-complete by establishing the non-existence of a certain type of polymorphism.

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arXiv2017

Extremal Bounds for Bootstrap Percolation in the Hypercube

The $r$-neighbour bootstrap percolation process on a graph $G$ starts with an initial set $A_0$ of "infected" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ eventually becomes infected, then we say that $A_0$ percolates. We prove a conjecture of Balogh and Bollobás which says that, for fixed $r$ and $d\to\infty$, every percolating set in the $d$-dimensional hypercube has cardinality at least $\frac{1+o(1)}{r}\binom{d}{r-1}$. We also prove an analogous result for multidimensional rectangular grids. Our proofs exploit a connection between bootstrap percolation and a related process, known as weak saturation. In addition, we improve on the best known upper bound for the minimum size of a percolating set in the hypercube. In particular, when $r=3$, we prove that the minimum cardinality of a percolating set in the $d$-dimensional hypercube is $\left\lceil\frac{d(d+3)}{6}\right\rceil+1$ for all $d\geq3$.

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arXiv2017

Supersaturation in Posets and Applications Involving the Container Method

We consider 'supersaturation' problems in partially ordered sets (posets) of the following form. Given a finite poset $P$ and an integer $m$ greater than the cardinality of the largest antichain in $P$, what is the minimum number of comparable pairs in a subset of $P$ of cardinality $m$? We provide a framework for obtaining lower bounds on this quantity based on counting comparable pairs relative to a random chain and apply this framework to obtain supersaturation results for three classical posets: the boolean lattice, the collection of subspaces of $\mathbb{F}_q^n$ ordered by set inclusion and the set of divisors of the square of a square-free integer under the 'divides' relation. The bound that we obtain for the boolean lattice can be viewed as an approximate version of a known theorem of Kleitman. In addition, we apply our supersaturation results to obtain (a) upper bounds on the number of antichains in these posets and (b) asymptotic bounds on the cardinality of the largest antichain in $p$-random subsets of these posets which hold with high probability (for $p$ in a certain range). The proofs of these results rely on a 'container-type' lemma for posets which generalises a result of Balogh, Mycroft and Treglown. We also state a number of open problems regarding supersaturation in posets and counting antichains.

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arXiv2016

Weak regularity and finitely forcible graph limits

Graphons are analytic objects representing limits of convergent sequences of graphs. Lovász and Szegedy conjectured that every finitely forcible graphon, i.e. any graphon determined by finitely many graph densities, has a simple structure. In particular, one of their conjectures would imply that every finitely forcible graphon has a weak $\varepsilon$-regular partition with the number of parts bounded by a polynomial in $\varepsilon^{-1}$. We construct a finitely forcible graphon $W$ such that the number of parts in any weak $\varepsilon$-regular partition of $W$ is at least exponential in $\varepsilon^{-2}/2^{5\log^*\varepsilon^{-2}}$. This bound almost matches the known upper bound for graphs and, in a certain sense, is the best possible for graphons.

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arXiv2016

A Dichotomy Theorem for Circular Colouring Reconfiguration

The "reconfiguration problem" for circular colourings asks, given two $(p,q)$-colourings $f$ and $g$ of a graph $G$, is it possible to transform $f$ into $g$ by changing the colour of one vertex at a time such that every intermediate mapping is a $(p,q)$-colouring? We show that this problem can be solved in polynomial time for $2\leq p/q <4$ and is PSPACE-complete for $p/q\geq 4$. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings.

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