A subset R of the vertex set of a graph $\Gamma $ is said to be $(\kappa ,\tau )$-regular if R induces a $\kappa $-regular subgraph and every vertex outside R is adjacent to exactly $\tau $ vertices in R. In particular, if R is a $(\kappa ,\tau )$-regular set of some Cayley graph on a finite group G, then R is called a $(\kappa ,\tau )$-regular set of G. Let H be a nontrivial normal subgroup of G, and $\kappa $ and $\tau $ a pair of integers satisfying $0\leq \kappa \leq |H|-1$, $1\leq \tau \leq |H|$ and $\gcd (2,|H|-1)\mid \kappa $. It is proved that (i) if $\tau $ is even, then H is a $(\kappa ,\tau )$-regular set of G; (ii) if $\tau $ is odd, then H is a $(\kappa ,\tau )$-regular set of G if and only if it is a $(0,1)$-regular set of G.

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. The magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth–Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behaviour is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines the magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

Negative dependence of sequences of random variables is often an interesting characteristic of their distribution, as well as a useful tool for studying various asymptotic results, including central limit theorems, Poisson approximations, the rate of increase of the maximum, and more. In the study of probability models of tournaments, negative dependence of participants’ outcomes arises naturally, with application to various asymptotic results. In particular, the property of negative orthant dependence was proved in several articles for different tournament models, with a special proof for each model. In this note we unify these results by proving a stronger property, negative association, a generalization leading to a very simple proof. We also present a natural example of a knockout tournament where the scores are negatively orthant dependent but not negatively associated. The proof requires a new result on a preservation property of negative orthant dependence that is of independent interest.

Given a family $\mathcal{F}$ of bipartite graphs, the Zarankiewicz number $z(m,n,\mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $\mathcal{F}$ as a subgraph (such $G$ is called $\mathcal{F}$-free). For $1\leq \beta \lt \alpha \lt 2$, a family $\mathcal{F}$ of bipartite graphs is $(\alpha,\beta )$-smooth if for some $\rho \gt 0$ and every $m\leq n$, $z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any $(\alpha,\beta )$-smooth family $\mathcal{F}$, there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large $n$, any $n$-vertex $\mathcal{F}\cup \{C_k\}$-free graph with minimum degree at least $\rho (\frac{2n}{5}+o(n))^{\alpha -1}$ is bipartite. In this paper, we strengthen their result by showing that for every real $\delta \gt 0$, there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large $n$, any $n$-vertex $\mathcal{F}\cup \{C_k\}$-free graph with minimum degree at least $\delta n^{\alpha -1}$ is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families $\mathcal{F}$ consisting of the single graph $K_{s,t}$ when $t\gg s$. We also prove an analogous result for $C_{2\ell }$-free graphs for every $\ell \geq 2$, which complements a result of Keevash, Sudakov and Verstraëte.

Let G be a finite transitive group on a set $\Omega $, let $\alpha \in \Omega $, and let $G_{\alpha }$ be the stabilizer of the point $\alpha $ in G. In this paper, we are interested in the proportion

$$ \begin{align*} \frac{|\{\omega\in \Omega\mid \omega \textrm{ lies in a }G_{\alpha}\textrm{-orbit of cardinality at most 2}\}|}{|\Omega|}, \end{align*} $$

that is, the proportion of elements of $\Omega $ lying in a suborbit of cardinality at most 2. We show that, if this proportion is greater than $5/6$, then each element of $\Omega $ lies in a suborbit of cardinality at most 2, and hence G is classified by a result of Bergman and Lenstra. We also classify the permutation groups attaining the bound $5/6$.

We use these results to answer a question concerning the enumeration of Cayley graphs. Given a transitive group G containing a regular subgroup R, we determine an upper bound on the number of Cayley graphs on R containing G in their automorphism groups.

We investigate quantum lens spaces, $C(L_q^{2n+1}(r;\underline {m}))$, introduced by Brzeziński and Szymański as graph $C^*$-algebras. We give a new description of $C(L_q^{2n+1}(r;\underline {m}))$ as graph $C^*$-algebras amending an error in the original paper by Brzeziński and Szymański. Furthermore, for $n\leq 3$, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r. This builds upon the work of Eilers, Restorff, Ruiz, and Sørensen.

The classic game of Battleship involves two players taking turns attempting to guess the positions of a fleet of vertically or horizontally positioned enemy ships hidden on a $10\times 10$ grid. One variant of this game, also referred to as Battleship Solitaire, Bimaru or Yubotu, considers the game with the inclusion of X-ray data, represented by knowledge of how many spots are occupied in each row and column in the enemy board. This paper considers the Battleship puzzle problem: the problem of reconstructing an enemy fleet from its X-ray data. We generate non-unique solutions to Battleship puzzles via certain reflection transformations akin to Ryser interchanges. Furthermore, we demonstrate that solutions of Battleship puzzles may be reliably obtained by searching for solutions of the associated classical binary discrete tomography problem which minimise the discrete Laplacian. We reformulate this optimisation problem as a quadratic unconstrained binary optimisation problem and approximate solutions via a simulated annealer, emphasising the future practical applicability of quantum annealers to solving discrete tomography problems with predefined structure.

We first establish a lower bound on the size and spectral radius of a graph G to guarantee that G contains a fractional perfect matching. Then, we determine an upper bound on the distance spectral radius of a graph G to ensure that G has a fractional perfect matching. Furthermore, we construct some extremal graphs to show all the bounds are best possible.

Under the assumption that sequences of graphs equipped with resistances, associated measures, walks and local times converge in a suitable Gromov-Hausdorff topology, we establish asymptotic bounds on the distribution of the $\varepsilon$-blanket times of the random walks in the sequence. The precise nature of these bounds ensures convergence of the $\varepsilon$-blanket times of the random walks if the $\varepsilon$-blanket time of the limiting diffusion is continuous at $\varepsilon$ with probability 1. This result enables us to prove annealed convergence in various examples of critical random graphs, including critical Galton-Watson trees and the Erdős-Rényi random graph in the critical window. We highlight that proving continuity of the $\varepsilon$-blanket time of the limiting diffusion relies on the scale invariance of a finite measure that gives rise to realizations of the limiting compact random metric space, and therefore we expect our results to hold for other examples of random graphs with a similar scale invariance property.

The book graph $B_n ^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$. Here we consider the natural off-diagonal variant $r(B_{cn} ^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$. In this more general setting, we show that an interesting dichotomy emerges: for very small $c$, a simple $k$-partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being $k$-partite, while, for $c$ bounded away from $0$, random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of $c$.

We give an example of an FIID vertex-labeling of ${\mathbb T}_3$ whose marginals are uniform on $[0,1]$, and if we delete the edges between those vertices whose labels are different, then some of the remaining clusters are infinite. We also show that no such process can be finitary.

Let $G=(V,E)$ be a countable graph. The Bunkbed graph of $G$ is the product graph $G \times K_2$, which has vertex set $V\times \{0,1\}$ with “horizontal” edges inherited from $G$ and additional “vertical” edges connecting $(w,0)$ and $(w,1)$ for each $w \in V$. Kasteleyn’s Bunkbed conjecture states that for each $u,v \in V$ and $p\in [0,1]$, the vertex $(u,0)$ is at least as likely to be connected to $(v,0)$ as to $(v,1)$ under Bernoulli-$p$ bond percolation on the bunkbed graph. We prove that the conjecture holds in the $p \uparrow 1$ limit in the sense that for each finite graph $G$ there exists $\varepsilon (G)\gt 0$ such that the bunkbed conjecture holds for $p \geqslant 1-\varepsilon (G)$.

An old conjecture of Erdős and McKay states that if all homogeneous sets in an $n$-vertex graph are of order $O(\!\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$. We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\!\log n)$, then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$.

Given a graphon $W$ and a finite simple graph $H$, with vertex set $V(H)$, denote by $X_n(H, W)$ the number of copies of $H$ in a $W$-random graph on $n$ vertices. The asymptotic distribution of $X_n(H, W)$ was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where $H$ is a clique. In this paper, we extend this result to any fixed graph $H$. Towards this we introduce a notion of $H$-regularity of graphons and show that if the graphon $W$ is not $H$-regular, then $X_n(H, W)$ has Gaussian fluctuations with scaling $n^{|V(H)|-\frac{1}{2}}$. On the other hand, if $W$ is $H$-regular, then the fluctuations are of order $n^{|V(H)|-1}$ and the limiting distribution of $X_n(H, W)$ can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centred chi-squared random variables with the weights determined by the spectral properties of a graphon derived from $W$. Our proofs use the asymptotic theory of generalised $U$-statistics developed by Janson and Nowicki [22]. We also investigate the structure of $H$-regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also $H$-regular graphons $W$ for which both the Gaussian or the non-Gaussian components are degenerate, that is, $X_n(H, W)$ has a degenerate limit even under the scaling $n^{|V(H)|-1}$. We give an example of this degeneracy with $H=K_{1, 3}$ (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.

For a graph G and a family of graphs $\mathcal {F}$, the Turán number ${\mathrm {ex}}(G,\mathcal {F})$ is the maximum number of edges an $\mathcal {F}$-free subgraph of G can have. We prove that ${\mathrm {ex}}(G,\mathcal {F})\ge {\mathrm {ex}}(K_r, \mathcal {F})$ if the chromatic number of G is r and $\mathcal {F}$ is a family of connected graphs. This result answers a question raised by Briggs and Cox [‘Inverting the Turán problem’, Discrete Math. 342(7) (2019), 1865–1884] about the inverse Turán number for all connected graphs.

We consider supercritical site percolation on the $d$-dimensional hypercube $Q^d$. We show that typically all components in the percolated hypercube, besides the giant, are of size $O(d)$. This resolves a conjecture of Bollobás, Kohayakawa, and Łuczak from 1994.

A $(p,q)$-colouring of a graph $G$ is an edge-colouring of $G$ which assigns at least $q$ colours to each $p$-clique. The problem of determining the minimum number of colours, $f(n,p,q)$, needed to give a $(p,q)$-colouring of the complete graph $K_n$ is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers $r_k(p)$. The best-known general upper bound on $f(n,p,q)$ was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where $p=q$ have been obtained only for $p\in \{4,5\}$, each of which was proved by giving a deterministic construction which combined a $(p,p-1)$-colouring using few colours with an algebraic colouring.

In this paper, we provide a framework for proving new upper bounds on $f(n,p,p)$ in the style of these earlier constructions. We characterize all colourings of $p$-cliques with $p-1$ colours which can appear in our modified version of the $(p,p-1)$-colouring of Conlon, Fox, Lee, and Sudakov. This allows us to greatly reduce the amount of case-checking required in identifying $(p,p)$-colourings, which would otherwise make this problem intractable for large values of $p$. In addition, we generalize our algebraic colouring from the $p=5$ setting and use this to give improved upper bounds on $f(n,6,6)$ and $f(n,8,8)$.

A recent paper of Balogh, Li and Treglown [3] initiated the study of Dirac-type problems for ordered graphs. In this paper, we prove a number of results in this area. In particular, we determine asymptotically the minimum degree threshold for forcing

1. (i) a perfect H-tiling in an ordered graph, for any fixed ordered graph H of interval chromatic number at least $3$;

2. (ii) an H-tiling in an ordered graph G covering a fixed proportion of the vertices of G (for any fixed ordered graph H);

3. (iii) an H-cover in an ordered graph (for any fixed ordered graph H).

The first two of these results resolve questions of Balogh, Li and Treglown, whilst (iii) resolves a question of Falgas-Ravry. Note that (i) combined with a result from [3] completely determines the asymptotic minimum degree threshold for forcing a perfect H-tiling. Additionally, we prove a result that, combined with a theorem of Balogh, Li and Treglown, asymptotically determines the minimum degree threshold for forcing an almost perfect H-tiling in an ordered graph (for any fixed ordered graph H). Our work therefore provides ordered graph analogues of the seminal tiling theorems of Kühn and Osthus [Combinatorica 2009] and of Komlós [Combinatorica 2000]. Each of our results exhibits some curious, and perhaps unexpected, behaviour. Our solution to (i) makes use of a novel absorbing argument.

In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our method to a model of a random intersection graph, a random graph obtained through p-bond percolation on a general d-regular graph, and a model of an inhomogeneous random graph.

We provide a general purpose result for the coupling of exploration processes of random graphs, both undirected and directed, with their local weak limits when this limit is a marked Galton–Watson process. This class includes in particular the configuration model and the family of inhomogeneous random graphs with rank-1 kernel. Vertices in the graph are allowed to have attributes on a general separable metric space and can potentially influence the construction of the graph itself. The coupling holds for any fixed depth of a breadth-first exploration process.