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Alex Scott

Publications and source records attributed to Alex Scott.

At least 55 records · Page 3Linked to original sources
arXiv2024

Distant digraph domination

A {\em $k$-kernel} in a digraph $G$ is a stable set $X$ of vertices such that every vertex of $G$ can be joined from $X$ by a directed path of length at most $k$. We prove three results about $k$-kernels. First, it was conjectured by Erdős and Székely in 1976 that every digraph $G$ with no source has a 2-kernel $|K|$ with $|K|\le |G|/2$. We prove this conjecture when $G$ is a ``split digraph'' (that is, its vertex set can be partitioned into a tournament and a stable set), improving a result of Langlois et al., who proved that every split digraph $G$ with no source has a 2-kernel of size at most $2|G|/3$. Second, the Erdős-Székely conjecture implies that in every digraph $G$ there is a 2-kernel $K$ such that the union of $K$ and its out-neighbours has size at least $|G|/2$. We prove that this is true if $V(G)$ can be partitioned into a tournament and an acyclic set. Third, in a recent paper, Spiro asked whether, for all $k\ge 3$, every strongly-connected digraph $G$ has a $k$-kernel of size at most about $|G|/(k+1)$. This remains open, but we prove that there is one of size at most about $|G|/(k-1)$.

math.CO↗
arXiv2024

Induced subgraph density. III. Cycles and subdivisions

We show that for every two cycles $C,D$, there exists $c>0$ such that if $G$ is both $C$-free and $\overline{D}$-free then $G$ has a clique or stable set of size at least $|G|^c$. ("$H$-free" means with no induced subgraph isomorphic to $H$, and $\overline{D}$ denotes the complement graph of $D$.) Since the five-vertex cycle $C_5$ is isomorphic to its complement, this extends the earlier result that $C_5$ satisfies the Erdős-Hajnal conjecture. It also unifies and strengthens several other results. The results for cycles are special cases of results for subdivisions, as follows. Let $H,J$ be obtained from smaller graphs by subdividing every edge exactly twice. We will prove that there exists $c>0$ such that if $G$ is both $H$-free and $\overline{J}$-free then $G$ has a clique or stable set of size at least $|G|^c$. And the same holds if $H$ and/or $J$ is obtained from a graph bychoosing a forest $F$ and subdividing every edge not in $F$ at least five times. Our proof uses the framework of iterative sparsification developed in other papers of this series. Along the way, we will also give a short and simple proof of a celebrated result of Fox and Sudakov, that says that for all $H$, every $H$-free graph contains either a large stable set or a large complete bipartite subgraph.

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arXiv2024

Lower bounds for graph reconstruction with maximal independent set queries

We investigate the number of maximal independent set queries required to reconstruct the edges of a hidden graph. We show that randomised adaptive algorithms need at least $Ω(Δ^2 \log(n / Δ) / \log Δ)$ queries to reconstruct $n$-vertex graphs of maximum degree $Δ$ with success probability at least $1/2$, and we further improve this lower bound to $Ω(Δ^2 \log(n / Δ))$ for randomised non-adaptive algorithms. We also prove that deterministic non-adaptive algorithms require at least $Ω(Δ^3 \log n / \log Δ)$ queries. This improves bounds of Konrad, O'Sullivan, and Traistaru, and answers one of their questions. The proof of the lower bound for deterministic non-adaptive algorithms relies on a connection to cover-free families, for which we also improve known bounds.

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arXiv2024

A note on graphs of $k$-colourings

For a graph $G$, the $k$-colouring graph of $G$ has vertices corresponding to proper $k$-colourings of $G$ and edges between colourings that differ at a single vertex. The graph supports the Glauber dynamics Markov chain for $k$-colourings, and has been extensively studied from both extremal and probabilistic perspectives. In this note, we show that for every graph $G$, there exists $k$ such that $G$ is uniquely determined by its $k$-colouring graph, confirming two conjectures of Asgarli, Krehbiel, Levinson and Russell. We further show that no finite family of generalised chromatic polynomials for $G$, which encode induced subgraph counts of its colouring graphs, uniquely determine $G$.

math.CO↗
arXiv2024

Induced subgraph density. I. A loglog step towards Erdos-Hajnal

In 1977, Erdős and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^c$; and they proved that this is true with $ |G|^c$ replaced by $2^{c\sqrt{\log |G|}}$. Until now, there has been no improvement on this result (for general $H$). We prove a strengthening: that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ with $|G|\ge 2$ has a clique or stable set of size at least $$2^{c\sqrt{\log |G|\log\log|G|}}.$$ Indeed, we prove the corresponding strengthening of a theorem of Fox and Sudakov, which in turn was a common strengthening of theorems of Rödl, Nikiforov, and the theorem of Erdős and Hajnal mentioned above.

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arXiv2024

Pure pairs. IX. Transversal trees

Fix k>0, and let G be a graph, with vertex set partitioned into k subsets (`blocks') of approximately equal size. An induced subgraph of G is transversal (with respect to this partition) if it has exactly one vertex in each block (and therefore it has exactly k vertices). A pure pair in G is a pair X,Y of disjoint subsets of V(G) such that either all edges between X,Y are present or none are; and in the present context we are interested in pure pairs (X,Y) where each of X,Y is a subset of one of the blocks, and not the same block. This paper collects several results and open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded.

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arXiv2024

Invertibility of digraphs and tournaments

For an oriented graph $D$ and a set $X\subseteq V(D)$, the inversion of $X$ in $D$ is the digraph obtained by reversing the orientations of the edges of $D$ with both endpoints in $X$. The inversion number of $D$, $\textrm{inv}(D)$, is the minimum number of inversions which can be applied in turn to $D$ to produce an acyclic digraph. Answering a recent question of Bang-Jensen, da Silva, and Havet we show that, for each $k\in\mathbb{N}$ and tournament $T$, the problem of deciding whether $\textrm{inv}(T)\leq k$ is solvable in time $O_k(|V(T)|^2)$, which is tight for all $k$. In particular, the problem is fixed-parameter tractable when parameterised by $k$. On the other hand, we build on their work to prove their conjecture that for $k\geq 1$ the problem of deciding whether a general oriented graph $D$ has $\textrm{inv}(D)\leq k$ is NP-complete. We also construct oriented graphs with inversion number equal to twice their cycle transversal number, confirming another conjecture of Bang-Jensen, da Silva, and Havet, and we provide a counterexample to their conjecture concerning the inversion number of so-called 'dijoin' digraphs while proving that it holds in certain cases. Finally, we asymptotically solve the natural extremal question in this setting, improving on previous bounds of Belkhechine, Bouaziz, Boudabbous, and Pouzet to show that the maximum inversion number of an $n$-vertex tournament is $(1+o(1))n$.

math.CO↗
arXiv2024

Reconstruction of shredded random matrices

A matrix is given in ``shredded'' form if we are presented with the multiset of rows and the multiset of columns, but not told which row is which or which column is which. The matrix is reconstructible if it is uniquely determined by this information. Let $M$ be a random binary $n\times n$ matrix, where each entry independently is $1$ with probability $p=p(n)\le\frac12$. Atamanchuk, Devroye and Vicenzo introduced the problem and showed that $M$ is reconstructible with high probability for $p\ge (2+\varepsilon)\frac{1}{n}\log n$. Here we find that the sharp threshold for reconstructibility is at $p\sim\frac{1}{2n}\log n$.

math.CO↗
arXiv2023

Reconstruction from smaller cards

The $\ell$-deck of a graph $G$ is the multiset of all induced subgraphs of $G$ on $\ell$ vertices. We say that a graph is reconstructible from its $\ell$-deck if no other graph has the same $\ell$-deck. In 1957, Kelly showed that every tree with $n\ge3$ vertices can be reconstructed from its $(n-1)$-deck, and Giles strengthened this in 1976, proving that trees on at least 6 vertices can be reconstructed from their $(n-2)$-decks. Our main theorem states that trees are reconstructible from their $(n-r)$-decks for all $r\le n/{9}+o(n)$, making substantial progress towards a conjecture of Nýdl from 1990. In addition, we can recognise the connectedness of a graph from its $\ell$-deck when $\ell\ge 9n/10$, and reconstruct the degree sequence when $\ell\ge\sqrt{2n\log(2n)}$. All of these results are significant improvements on previous bounds.

math.CO↗
arXiv2023

Graphs with arbitrary Ramsey number and connectivity

The Ramsey number $r(G)$ of a graph $G$ is the minimum number $N$ such that any red-blue colouring of the edges of $K_N$ contains a monochromatic copy of $G$. Pavez-Signé, Piga and Sanhueza-Matamala proved that for any function $n\leq f(n) \leq r(K_n)$, there is a sequence of connected graphs $(G_n)_{n\in \mathbb{N}}$ with $|V(G_n)|=n$ such that $r(G_n)=Θ(f(n))$ and conjectured that $G_n$ can additionally have arbitrarily large connectivity. In this note we prove their conjecture.

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arXiv2023

Induced $C_4$-free subgraphs with large average degree

We prove that there exists a constant $C$ so that, for all $s,k \in \mathbb{N}$, if $G$ has average degree at least $k^{Cs^3}$ and does not contain $K_{s,s}$ as a subgraph then it contains an induced subgraph which is $C_4$-free and has average degree at least $k$. It was known that some function of $s$ and $k$ suffices, but this is the first explicit bound. We give several applications of this result, including short and streamlined proofs of the following two corollaries. We show that there exists a constant $C$ so that, for all $s,k \in \mathbb{N}$, if $G$ has average degree at least $k^{Cs^3}$ and does not contain $K_{s,s}$ as a subgraph then it contains an induced subdivision of $K_k$. This is the first quantitative improvement on a well-known theorem of Kühn and Osthus; their proof gives a bound that is triply exponential in both $k$ and $s$. We also show that for any hereditary degree-bounded class $\mathcal{F}$, there exists a constant $C=C_\mathcal{F}$ so that $C^{s^3}$ is a degree-bounding function for $\mathcal{F}$. This is the first bound of any type on the rate of growth of such functions. It is open whether there is always a polynomial degree-bounding function.

math.CO↗
arXiv2023

Pure pairs. VIII. Excluding a sparse graph

A pure pair of size $t$ in a graph $G$ is a pair $A,B$ of disjoint sets of $t$ vertices such that $A$ is either complete or anticomplete to $B$. It is known that, for every forest $H$, every graph on $n\ge2$ vertices that does not contain $H$ or its complement as an induced subgraph has a pure pair of size $Ω(n)$; furthermore, this only holds when $H$ or its complement is a forest. In this paper, we look at pure pairs of size $n^{1-c}$, where $0 c$.

math.CO↗
arXiv2023

Induced subgraphs of induced subgraphs of large chromatic number

We prove that, for every graph $F$ with at least one edge, there is a constant $c_F$ such that there are graphs of arbitrarily large chromatic number and the same clique number as $F$ in which every $F$-free induced subgraph has chromatic number at most $c_F$. This generalises recent theorems of Briański, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Our results imply that for every $r\geq 3$ the class of $K_r$-free graphs has a very strong vertex Ramsey-type property, giving a vast generalisation of a result of Folkman from 1970. We also prove related results for tournaments, hypergraphs and infinite families of graphs, and show an analogous statement for graphs where clique number is replaced by odd girth.

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arXiv2023

Boundary rigidity of 3D CAT(0) cube complexes

The boundary rigidity problem is a classical question from Riemannian geometry: if $(M, g)$ is a Riemannian manifold with smooth boundary, is the geometry of $M$ determined up to isometry by the metric $d_g$ induced on the boundary $\partial M$? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than $4$. We prove a $3$-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in $\mathbb{R}^3$ can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave's result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.

math.CO↗
arXiv2023

Pure pairs. X. Tournaments and the strong Erdos-Hajnal property

A pure pair in a tournament $G$ is an ordered pair $(A,B)$ of disjoint subsets of $V(G)$ such that every vertex in $B$ is adjacent from every vertex in $A$. Which tournaments $H$ have the property that if $G$ is a tournament not containing $H$ as a subtournament, and $|G|>1$, there is a pure pair $(A,B)$ in $G$ with $|A|,|B|\ge c|G|$, where $c>0$ is a constant independent of $G$? Let us say that such a tournament $H$ has the strong EH-property. As far as we know, it might be that a tournament $H$ has this property if and only if its vertex set has a linear ordering in which its backedges form a forest. Certainly this condition is necessary, but we are far from proving sufficiency. We make a small step in this direction, showing that if a tournament can be ordered with at most three backedges then it has the strong EH-property (except for one case, that we could not decide). In particular, every tournament with at most six vertices has the property, except for three that we could not decide. We also give a seven-vertex tournament that does not have the strong EH-property. This is related to the Erdos-Hajnal conjecture, which in one form says that for every tournament $H$ there exists $τ>0$ such that every tournament $G$ not containing $H$ as a subtournament has a transitive subtournament of cardinality at least $|G|^τ$. Let us say that a tournament $H$ satisfying this has the EH-property. It is known that every tournament with the strong EH-property also has the EH-property; so our result extends work by Berger, Choromanski and Chudnovsky, who proved that every tournament with at most six vertices has the EH-property, except for one that they did not decide.

math.CO↗
arXiv2023

The structure and density of $k$-product-free sets in the free semigroup

The free semigroup $\mathcal{F}$ over a finite alphabet $\mathcal{A}$ is the set of all finite words with letters from $\mathcal{A}$ equipped with the operation of concatenation. A subset $S$ of $\mathcal{F}$ is $k$-product-free if no element of $S$ can be obtained by concatenating $k$ words from $S$, and strongly $k$-product-free if no element of $S$ is a (non-trivial) concatenation of at most $k$ words from $S$. We prove that a $k$-product-free subset of $\mathcal{F}$ has upper Banach density at most $1/ρ(k)$, where $ρ(k) = \min\{\ell \colon \ell \nmid k - 1\}$. We also determine the structure of the extremal $k$-product-free subsets for all $k \notin \{3, 5, 7, 13\}$; a special case of this proves a conjecture of Leader, Letzter, Narayanan, and Walters. We further determine the structure of all strongly $k$-product-free sets with maximum density. Finally, we prove that $k$-product-free subsets of the free group have upper Banach density at most $1/ρ(k)$, which confirms a conjecture of Ortega, Rué, and Serra.

math.CO↗
arXiv2023

Flashes and rainbows in tournaments

Colour the edges of the complete graph with vertex set $\{1, 2, \dotsc, n\}$ with an arbitrary number of colours. What is the smallest integer $f(l,k)$ such that if $n > f(l,k)$ then there must exist a monotone monochromatic path of length $l$ or a monotone rainbow path of length $k$? Lefmann, Rödl, and Thomas conjectured in 1992 that $f(l, k) = l^{k - 1}$ and proved this for $l \ge (3 k)^{2 k}$. We prove the conjecture for $l \geq k^3 (\log k)^{1 + o(1)}$ and establish the general upper bound $f(l, k) \leq k (\log k)^{1 + o(1)} \cdot l^{k - 1}$. This reduces the gap between the best lower and upper bounds from exponential to polynomial in $k$. We also generalise some of these results to the tournament setting.

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