arXiv subjects

Alex Scott

Publications and source records attributed to Alex Scott.

At least 91 records · Page 5Linked to original sources

arXiv2022

Strengthening Rodl's theorem

What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rodl showed in the 1980s that every H-free graph has large parts that are very dense or very sparse. More precisely, let us say that a graph F on n vertices is c-restricted if either F or its complement has maximum degree at most cn. Rodl proved that for every graph H, and every c>0, every H-free graph G has a linear-sized set of vertices inducing a c-restricted graph. We strengthen Rodl's result as follows: for every graph H, and all c>0, every H-free graph can be partitioned into a bounded number of subsets inducing c-restricted graphs.

math.CO↗

arXiv2022

A note on infinite antichain density

Let $\mathcal{F}$ be an antichain of finite subsets of $\mathbb{N}$. How quickly can the quantities $|\mathcal{F}\cap 2^{[n]}|$ grow as $n\to\infty$? We show that for any sequence $(f_n)_{n\ge n_0}$ of positive integers satisfying $\sum_{n=n_0}^\infty f_n/2^n \le 1/4$, $f_{n_0}=1$ and $f_n\le f_{n+1}\le 2f_n$, there exists an infinite antichain $\mathcal{F}$ of finite subsets of $\mathbb{N}$ such that $|\mathcal{F}\cap 2^{[n]}| \geq f_n$ for all $n\ge n_0$. It follows that for any $\varepsilon>0$ there exists an antichain $\mathcal{F}\subseteq 2^\mathbb{N}$ such that $$\liminf_{n \to \infty} |\mathcal{F}\cap 2^{[n]}| \cdot \left(\frac{2^n}{n\log^{1+\varepsilon} n}\right)^{-1} > 0.$$ This resolves a problem of Sudakov, Tomon and Wagner in a strong form, and is essentially tight.

math.CO↗

arXiv2022

Balancing connected colourings of graphs

We show that the edges of any graph $G$ containing two edge-disjoint spanning trees can be blue/red coloured so that the blue and red graphs are connected and the blue and red degrees at each vertex differ by at most four. This improves a result of Hörsch. We discuss variations of the question for digraphs, infinite graphs and a computational question, and resolve two further questions of Hörsch in the negative.

math.CO↗

arXiv2022

Bipartite graphs with no $K_6$ minor

A theorem of Mader shows that every graph with average degree at least eight has a $K_6$ minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do not know whether all graphs with minimum degree at least seven have $K_6$ minors, but minimum degree six is certainly not enough. For every $c>0$ there are arbitrarily large graphs with average degree at least $8-c$ and minimum degree at least six, with no $K_6$ minor. But what if we restrict ourselves to bipartite graphs? The first statement remains true: for every $c>0$ there are arbitrarily large bipartite graphs with average degree at least $8-c$ and no $K_6$ minor. But surprisingly, going to minimum degree now makes a significant difference. We will show that every bipartite graph with minimum degree at least six has a $K_6$ minor. Indeed, it is enough that every vertex in the larger part of the bipartition has degree at least six.

math.CO↗

arXiv2022

Polynomial bounds for chromatic number VII. Disjoint holes

A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to its clique number. In this paper we show that, for any $k$, if $G$ does not contain a $k$-multihole, then its chromatic number is at most a polynomial function of its clique number. We show that the same result holds if we ask for all the holes to be odd or of length four; and if we ask for the holes to be longer than any fixed constant or of length four. This is part of a broader study of graph classes that are polynomially $χ$-bounded.

math.CO↗

arXiv2022

Clustered colouring of graph classes with bounded treedepth or pathwidth

The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We determine the clustered chromatic number of any minor-closed class with bounded treedepth, and prove a best possible upper bound on the clustered chromatic number of any minor-closed class with bounded pathwidth. As a consequence, we determine the fractional clustered chromatic number of every minor-closed class.

math.CO↗

arXiv2021

Exact stability for Turán's Theorem

Turán's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability Theorem of Erdős and Simonovits shows that if a $K_{r+1}$-free graph with $n$ vertices has close to the maximal $t_r(n)$ edges, then it is close to being $r$-partite. In this paper we determine exactly the $K_{r+1}$-free graphs with at least $m$ edges that are farthest from being $r$-partite, for any $m\ge t_r(n) - δ_r n^2$. This extends work by Erdős, Győri and Simonovits, and proves a conjecture of Balogh, Clemen, Lavrov, Lidický and Pfender.

math.CO↗

arXiv2021

Lipschitz bijections between boolean functions

We answer four questions from a recent paper of Rao and Shinkar on Lipschitz bijections between functions from $\{0,1\}^n$ to $\{0,1\}$. (1) We show that there is no $O(1)$-bi-Lipschitz bijection from $\mathrm{Dictator}$ to $\mathrm{XOR}$ such that each output bit depends on $O(1)$ input bits. (2) We give a construction for a mapping from $\mathrm{XOR}$ to $\mathrm{Majority}$ which has average stretch $O(\sqrt{n})$, matching a previously known lower bound. (3) We give a 3-Lipschitz embedding $ϕ: \{0,1\}^n \to \{0,1\}^{2n+1}$ such that $\mathrm{XOR}(x) = \mathrm{Majority}(ϕ(x))$ for all $x \in \{0,1\}^n$. (4) We show that with high probability there is a $O(1)$-bi-Lipschitz mapping from $\mathrm{Dictator}$ to a uniformly random balanced function.

math.CO↗

arXiv2021

Active clustering for labeling training data

Gathering training data is a key step of any supervised learning task, and it is both critical and expensive. Critical, because the quantity and quality of the training data has a high impact on the performance of the learned function. Expensive, because most practical cases rely on humans-in-the-loop to label the data. The process of determining the correct labels is much more expensive than comparing two items to see whether they belong to the same class. Thus motivated, we propose a setting for training data gathering where the human experts perform the comparatively cheap task of answering pairwise queries, and the computer groups the items into classes (which can be labeled cheaply at the very end of the process). Given the items, we consider two random models for the classes: one where the set partition they form is drawn uniformly, the other one where each item chooses its class independently following a fixed distribution. In the first model, we characterize the algorithms that minimize the average number of queries required to cluster the items and analyze their complexity. In the second model, we analyze a specific algorithm family, propose as a conjecture that they reach the minimum average number of queries and compare their performance to a random approach. We also propose solutions to handle errors or inconsistencies in the experts' answers.

cs.DS↗

arXiv2021

A note on simplicial cliques

Motivated by an application in condensed matter physics and quantum information theory, we prove that every non-null even-hole-free claw-free graph has a simplicial clique, that is, a clique $K$ such that for every vertex $v \in K$, the set of neighbours of $v$ outside of $K$ is a clique. In fact, we prove the existence of a simplicial clique in a more general class of graphs defined by forbidden induced subgraphs.

math.CO↗

arXiv2021

Parking on the integers

Models of parking in which cars are placed randomly and then move according to a deterministic rule have been studied since the work of Konheim and Weiss in the 1960s. Recently, Damron, Gravner, Junge, Lyu, and Sivakoff introduced a model in which cars are both placed and move at random. Independently at each point of a Cayley graph $G$, we place a car with probability $p$, and otherwise an empty parking space. Each car independently executes a random walk until it finds an empty space in which to park. In this paper we introduce three new techniques for studying the model, namely the space-based parking model, and the strategies for parking and for car removal. These allow us to study the original model by coupling it with models where parking behaviour is easier to control. Applying our methods to the one-dimensional parking problem in $\mathbb{Z}$, we improve on previous work, showing that for $p<1/2$ the expected journey length of a car is finite, and for $p=1/2$ the expected journey length by time $t$ grows like $t^{3/4}$ up to a polylogarithmic factor.

math.PR↗

arXiv2021

Polynomial bounds for chromatic number. III. Excluding a double star

A double star is a tree with two internal vertices. It is known that the Gyárfás-Sumner conjecture holds for double stars, that is, for every double star $H$, there is a function $f$ such that if $G$ does not contain $H$ as an induced subgraph then $χ(G)\le f(ω(G))$ (where $χ, ω$ are the chromatic number and the clique number of $G$). Here we prove that $f$ can be chosen to be a polynomial.

math.CO↗

arXiv2021

Polynomial bounds for chromatic number. I. Excluding a biclique and an induced tree

Let H be a tree. It was proved by Rodl that graphs that do not contain H as an induced subgraph, and do not contain the complete bipartite graph $K_{t,t}$ as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened this, showing that such graphs have bounded degeneracy. Here we give a further strengthening, proving that for every tree H, the degeneracy is at most polynomial in t. This answers a question of Bonamy, Pilipczuk, Rzazewski, Thomasse and Walczak.

math.CO↗

arXiv2021

Polynomial bounds for chromatic number. II. Excluding a star-forest

The Gyarfas-Sumner conjecture says that for every forest $H$, there is a function $f$ such that if $G$ is $H$-free then $χ(G)\le f(ω(G))$ (where $χ, ω$ are the chromatic number and the clique number of $G$). Louis Esperet conjectured that, whenever such a statement holds, $f$ can be chosen to be a polynomial. The Gyarfas-Sumner conjecture is only known to be true for a modest set of forests $H$, and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known when $H$ is a five-vertex path. Here we prove Esperet's conjecture when each component of $H$ is a star.

math.CO↗

arXiv2021

Optimal labelling schemes for adjacency, comparability, and reachability

We construct asymptotically optimal adjacency labelling schemes for every hereditary class containing $2^{Ω(n^2)}$ $n$-vertex graphs as $n\to \infty$. This regime contains many classes of interest, for instance perfect graphs or comparability graphs, for which we obtain an adjacency labelling scheme with labels of $n/4+o(n)$ bits per vertex. This implies the existence of a reachability labelling scheme for digraphs with labels of $n/4+o(n)$ bits per vertex and comparability labelling scheme for posets with labels of $n/4+o(n)$ bits per element. All these results are best possible, up to the lower order term.

math.CO↗

arXiv2021

Powers of paths and cycles in tournaments

We show that for every positive integer $k$, any tournament can be partitioned into at most $2^{ck}$ $k$-th powers of paths. This result is tight up to the exponential constant. Moreover, we prove that for every $\varepsilon>0$ and every integer $k$, any tournament on $n\ge \varepsilon^{-Ck}$ vertices which is $\varepsilon$-far from being transitive contains the $k$-th power of a cycle of length $Ω(\varepsilon n)$; both bounds are tight up to the implied constants.

math.CO↗

arXiv2021

Induced subgraphs of graphs with large chromatic number. V. Chandeliers and strings

It is known that every graph of sufficiently large chromatic number and bounded clique number contains, as an induced subgraph, a subdivision of any fixed forest, and a subdivision of any fixed cycle. Equivalently, forests and triangles are pervasive, where H is pervasive (in some class of graphs) if for all s>0, every graph in the class with bounded clique number and sufficiently large chromatic number contains an induced subdivision of H, with every edge subdivided at least s times. Which other graphs are pervasive? Chalopin, Esperet, Li and Ossona de Mendez proved that every such graph is a forest of lanterns: roughly, the blocks are lanterns (graphs obtained from a tree by adding one extra vertex), and there are rules about how blocks fit together. It is not known whether every forest of lanterns is pervasive; but in another paper two of us prove that banana trees (multigraphs obtained from a forest by adding parallel edges) are pervasive, thus generalizing the two results above. This paper contains the first half of the proof, which works for any forest of lanterns, not just for banana trees. A class of graphs is r-controlled if for every graph in the class, its chromatic number is at most some function (determined by the class) of the largest chromatic number of an r-ball in the graph. In this paper we prove that for all r>1, every forest of lanterns is pervasive in every r-controlled class These results turn out particularly nicely when applied to string graphs (intersection graphs of sets of curves in the plane). A chandelier is a graph obtained from a tree by adding a vertex adjacent to its leaves. We prove that the class of string graphs is 2-controlled, and thus forests of lanterns are pervasive in this class. Furthermore, string graphs of sufficiently large chromatic number and bounded clique number contain any fixed chandelier as an induced subgraph.

math.CO↗

arXiv2021

Pure pairs. VI. Excluding an ordered tree

A pure pair in a graph $G$ is a pair $(Z_1,Z_2)$ of disjoint sets of vertices such that either every vertex in $Z_1$ is adjacent to every vertex in $Z_2$, or there are no edges between $Z_1$ and $Z_2$. With Maria Chudnovsky, we recently proved that, for every forest $F$, every graph $G$ with at least two vertices that does not contain $F$ or its complement as an induced subgraph has a pure pair $(Z_1,Z_2)$ with $|Z_1|,|Z_2|$ linear in $|G|$. Here we investigate what we can say about pure pairs in an {\em ordered} graph $G$, when we exclude an ordered forest $F$ and its complement as induced subgraphs. Fox showed that there need not be a linear pure pair; but Pach and Tomon showed that if $F$ is a monotone path then there is a pure pair of size $c|G|/\log |G|$. We generalise this to all ordered forests, at the cost of a slightly worse bound: we prove that, for every ordered forest $F$, every ordered graph $G$ with at least two vertices that does not contain $F$ or its complement as an induced subgraph has a pure pair of size $|G|^{1-o(1)}$.

math.CO↗