Let \(\mu (\varepsilon )\) be the minimum number of cubes of side \(\varepsilon \) needed to cover the unit three-torus \([\mathbb {R}/\mathbb {Z}]^3\). We prove new lower and upper bounds for \(\mu (\varepsilon )\) and find the exact value for all \(\varepsilon \ge \frac{7}{15}\) and all \(\varepsilon \in \left[ \frac{1}{r+1/(r^2+r+1)},\frac{r^2-1}{r^3-r-1}\right) \) for any integer \(r\ge 3\).

We prove that the family of largest cuts in the binomial random graph exhibits the following stability property: If $1/n \ll p = 1-\Omega(1)$, then, with high probability, there is a set of $n - o(n)$ vertices that is partitioned in the same manner by all maximum cuts of $G_{n,p}$. Moreover, the analogous statement remains true when one replaces maximum cuts with nearly-maximum cuts. We then demonstrate how one can use this statement as a tool for showing that certain properties of $G_{n,p}$ that hold in a fixed balanced cut hold simultaneously in all maximum cuts. We provide two example applications of this tool. First, we prove that maximum cuts in $G_{n,p}$ typically partition the neighbourhood of every vertex into nearly equal parts; this resolves a conjecture of DeMarco and Kahn for all but a narrow range of densities $p$. Second, for all edge-critical, nonbipartite, and strictly 2-balanced graphs $H$, we prove a lower bound on the threshold density $p$ above which every largest $H$-free subgraph of $G_{n,p}$ is $(\chi(H)-1)$-partite. Our lower bound exactly matches the upper bound on this threshold recently obtained by the first two authors.

Asymptotic behaviour of maximum sizes of induced trees and forests has been studied extensively in last decades, though the overall picture is far from being complete. In this paper, we close several significant gaps: 1) We prove 2-point concentration of the maximum sizes of an induced forest and an induced tree with maximum degree at most Δ in dense binomial random graphs G(n,p) with constant probability p. 2) We show concentration in an explicit interval of size o(1/p) for the maximum size of an induced forest with maximum degree at most Δ for 1/n≪ p=o(1). Our proofs rely on both the second moment approach, with the probabilistic part involving Talagrand's concentration inequality and the analytical part involving saddle-point analysis, and new results on enumeration of labelled trees and forests that might be of their own interest.

We study what deterministic distributed algorithms can compute on random input graphs in extremely weak models of distributed computing: all nodes are anonymous, and in each communication round, nodes broadcast a message to all their neighbors, receive a (multi)set of messages from their neighbors, and update their local state. These correspond to the SB and MB models introduced by Hella et al. [PODC 2012] and are strictly weaker than the standard port-numbering PN and LOCAL models. We investigate what can be computed almost surely on random input graphs. We give a one-round deterministic SB-algorithm using O(log n)-bit messages that computes unique identifiers with high probability on anonymous networks sampled from G(n,p), where n^ε-1≤ p ≤ 1/2 and ε>0 is an arbitrarily small constant. This algorithm is inspired by canonical labeling techniques in graph isomorphism testing and can be used to "anonymize" existing distributed graph algorithms designed for the broadcast CONGEST and LOCAL models. In particular, we give a new anonymous algorithm that finds a triangle in O(1/ε) rounds on the above input distribution. We also investigate computational power of natural analogs of "Monte Carlo" and "Las Vegas" distributed graph algorithms in the random graph setting, and establish some new collapse and hierarchy results. For example, our work shows the collapse of the weak model hierarchy of Hella et al. on G(n,p), as apart from a vanishingly small fraction of input graphs, the SB model is as powerful as LOCAL.

We develop a framework for proving universality results in sparse random graphs. As a first application, we show that there exists an absolute constant C>1 such that, with high probability, for every fixed constant Δ, the binomial random graph G(n,Cln n/n) contains every n-vertex tree with maximum degree at most Δ. This answers a question of Montgomery (Advances in Mathematics, 2019). We also determine, for every p satisfying Cln n/n≤ p=n^-1+o(1), the minimum girth ℓ (up to an absolute multiplicative constant) for which with high probability G(n,p) contains all cycle factors of girth at least Ω(ℓ). In particular, with high probability G(n,Cln n/n) contains all cycle factors of girth at least 100ln n/lnln n, which is optimal up to a constant factor. This extends an earlier result of Ferber, Kronenberg, and Luh (Transaction of the American Mathematical Society, 2019) and significantly improves a corollary of a deep result of Kahn, Lubetzky, and Wormald (Communications on Pure and Applied Mathematics, 2017). One of the key ingredients in the proofs is establishing the optimal depth of linking systems in sparse random graphs.

We prove that whenever d=d(n)→∞ and n-d→∞ as n→∞, then with high probability for any non-trivial initial colouring, the colour refinement algorithm distinguishes all vertices of the random regular graph 𝒢_n,d. This, in particular, implies that with high probability 𝒢_n,d admits a canonical labelling computable in time O(min{n^ω,nd^2+ndlog n}), where ω<2.372 is the matrix multiplication exponent.

What is the maximum length f_max(ℓ , Σ ) of a facial cycle of an inclusion-maximal graph with girth at least ℓ embedded on a given surface Σ ? If Σ =𝒫 is a plane, we show that 3ℓ -11≤f_max(ℓ , 𝒫)≤ 8ℓ -13 . We also prove that f_max(ℓ , Σ ) is bounded for any integer ℓ and any closed surface Σ . For a fixed Σ , we show that Ω (ℓ ) =f_max(ℓ , Σ ) = O(ℓ ^2) , while for a fixed ℓ≥ 6 , f_max(ℓ , Σ )=Θ (g) , where g is the genus of Σ .

In graph bootstrap percolation, edges of an Erdős-Rényi random graph 𝒢_n,p are initially active. Activation spreads to other edges of the complete graph K_n by an iterative process governed by a fixed graph H, whereby an edge becomes active whenever it is the only inactive edge in a copy of H. If all edges of K_n are eventually activated, we say the process H-percolates. The case H=K_3 corresponds to the classical sharp threshold for connectivity in 𝒢_n,p. When H=K_4, there are close connections with 2-neighbor bootstrap percolation from statistical physics. Varying H produces a wide range of behaviors. In this work, for every graph H, we locate the critical H-percolation threshold p_c(n,H), answering a question of Balogh, Bollobás, and Morris. Our general methods recover and improve several previous results. The location of p_c(n,H) is related to a critical limiting density ρ(H) of graphs that most efficiently activate a given edge. Introducing the parameter ρ(H) raises several questions. For instance, it remains open whether ρ(H) is computable in general, and its expression appears to indicate when the H-percolation threshold is sharp.

We show that there exist constants delta(1), delta(2) > 0 such that if G is an (n, d, lambda)-graph with lambda/d <= delta(1), then G contains an induced cycle of length at least delta 2n/d. We further demonstrate that, up to a constant factor, this is best possible. Utilising our techniques, we derive that the number of non-isomorphic induced subgraphs of such G is at least exponential in n log d/d, and further demonstrate that this is tight up to a constant factor in the exponent.

Concentration results say that a sequence of random variables becomes progressively concentrated around the mean. Such results are common in the study of functions of random graphs. We introduce a real-valued logic with various aggregate operators on graphs, including summation, and prove that every term in the language, seen as a random variable on random graphs within the classical Erdős-Rényi random graph model, is concentrated. We prove this for dense and sparse variants of Erdős-Rényi graphs. On the one hand, our results extend the line of work originating with Fagin and Glebskii et al. on zero-one laws for dense random graphs, as well as the zero-one law of Shelah and Spencer for sparse random graphs. On the other hand, they can be seen as a meta-theorem for inferring concentration results on random graphs, and we give examples of such applications.

Let X_H denote the number of copies of a fixed graph H in G_n, p. Gilmer and Kopparty conjectured that X_H satisfies a local central limit theorem (LCLT) provided that H is connected, p ≫ n^-1/m(H), and n^2 (1-p) ≫ 1, where m(H) is the maximum density. Following the work of Berkowitz, Sah and Sawhney confirmed this conjecture for every constant p, leaving the regime where p=o(1) open. In this regime, the only case addressed in the literature is when H=K_3, where, in a recent paper, Araújo and Mattos confirmed the conjecture for p ∈ (4n^-1/2, 1/2). This, together with a general result of Röllin and Ross, essentially settles the conjecture for the triangle. We generalise these results by showing that an LCLT holds for H = K_r (for any fixed r ≥ 3) in the regime n^-1/m(H)≪ p≤ 1/2, essentially settling the conjecture for cliques.

For an arbitrary finite tree T, we find the exact value of the wort-case stabilisation time of majority dynamics on T. We also prove that for a perfect rooted cubic tree T with diameter D and uniformly random initial opinions, the dynamics stabilises in time τ∈(D/4,D/3) with high probability.

The weak saturation number $\mathrm{wsat}(n,F)$ is the minimum number of edges in a graph on $n$ vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of $F$. A usual approach to prove a lower bound for the weak saturation number is algebraic: if it is possible to embed edges of $K_n$ in a vector space in a certain way (depending on $F$), then the dimension of the subspace spanned by the images of the edges of $K_n$ is a lower bound for the weak saturation number. In this paper, we present a new combinatorial approach to prove lower bounds for weak saturation numbers that allows to establish worst-case tight (up to constant additive terms) general lower bounds as well as to get exact values of the weak saturation numbers for certain graph families. It is known (Alon, 1985) that, for every $F$, there exists $c_F$ such that $\mathrm{wsat}(n,F)=c_Fn(1+o(1))$. Our lower bounds imply that all values in the interval $\left[\frac{\delta}{2}-\frac{1}{\delta+1},\delta-1\right]$ with step size $\frac{1}{\delta+1}$ are achievable by $c_F$ (while any value outside this interval is not achievable).

For a given graph F, the F-saturation number of a graph G, denoted by sat(G,F), is the minimum number of edges in an edge-maximal F-free subgraph of G. In 2017, Korándi and Sudakov determined Image 1 asymptotically, where Image 2 denotes the Erdős–Rényi random graph and Kr is the complete graph on r vertices. In this paper, among other results, we present an asymptotic upper bound on Image 3 for any bipartite graph F and also an asymptotic lower bound on Image 3 for any complete bipartite graph F.

We consider bond percolation on the d-dimensional binary hypercube with p=c/d for fixed c>1. We prove that the typical diameter of the giant component L_1 is of order Θ(d), and the typical mixing time of the lazy random walk on L_1 is of order Θ(d^2). This resolves long-standing open problems of Bollobás, Kohayakawa and Łuczak from 1994, and of Benjamini and Mossel from 2003. A key component in our approach is a new tight large deviation estimate on the number of vertices in L_1 whose proof includes several novel ingredients: a structural description of the residue outside the giant component after sprinkling, a tight quantitative estimate on the spread of the giant in the hypercube, and a stability principle which rules out the disintegration of large connected sets under thinning. This toolkit further allows us to obtain optimal bounds on the expansion in L_1.

We show that for every epsilon > 0 there exists a sufficiently large d(0) is an element of N such that for every d >= d(0), whp the random d-regular graph G(n, d) contains a T-factor for every tree T on at most (1 - epsilon) d/ log d vertices. This is best possible since, for large enough integer d, whp G(n, d) does not contain a (1+epsilon)(d)/log d-star-factor. Our method gives a randomised algorithm which whp finds said T-factor and whose expected running time is O(n(1+o(1))), as well as an efficient deterministic counterpart.

Let $K<^>r_n$ be the complete $r$ -uniform hypergraph on $n$ vertices, that is, the hypergraph whose vertex set is $[n] \, :\! = \{1,2,\ldots ,n\}$ and whose edge set is $\binom {[n]}{r}$ . We form $G<^>r(n,p)$ by retaining each edge of $K<^>r_n$ independently with probability $p$ . An $r$ -uniform hypergraph $H\subseteq G$ is $F$ -saturated if $H$ does not contain any copy of $F$ , but any missing edge of $H$ in $G$ creates a copy of $F$ . Furthermore, we say that $H$ is weakly $F$ -saturated in $G$ if $H$ does not contain any copy of $F$ , but the missing edges of $H$ in $G$ can be added back one-by-one, in some order, such that every edge creates a new copy of $F$ . The smallest number of edges in an $F$ -saturated hypergraph in $G$ is denoted by ${\textit {sat}}(G,F)$ , and in a weakly $F$ -saturated hypergraph in $G$ by $\mathop {\mbox{$w$-${sat}$}}\! (G,F)$ . In 2017, Kor & aacute;ndi and Sudakov initiated the study of saturation in random graphs, showing that for constant $p$ , with high probability ${\textit {sat}}(G(n,p),K_s)=(1+o(1))n\log _{\frac {1}{1-p}}n$ , and $\mathop {\mbox{$w$-${sat}$}}\! (G(n,p),K_s)=\mathop {\mbox{$w$-${sat}$}}\! (K_n,K_s)$ . Generalising their results, in this paper, we solve the saturation problem for random hypergraphs $G<^>r(n,p)$ for cliques $K_s<^>r$ , for every $2\le r \lt s$ and constant $p$ .

We prove that (1+o(1))√(e/n) is the sharp threshold for the appearance of the square of a Hamilton cycle in G(n,p), confirming the conjecture of Kahn, Narayanan, and Park. We also find the exact asymptotics of the threshold for the emergence of a spanning subgraph isomorphic to a fixed graph F for a wide family of d-regular graphs F. This family includes almost all d-regular graphs.

We estimate the minimum number of distance queries that is sufficient to reconstruct the binomial random graph G(n,p) with constant diameter with high probability. We get a tight (up to a constant factor) answer for all p > n(-1+o(1)) outside "threshold windows" around n(-k/(k+1)+o(1)), k is an element of Z(>)o: with high probability the query complexity equals Theta(n(4-d)p(2-d)), where d is the diameter of the random graph. This demonstrates the following non-monotone behaviour: the query complexity jumps down at moments when the diameter gets larger; yet, between these moments the query complexity grows. We also show that there exists a non-adaptive algorithm that reconstructs the random graph with Theta(n(4-d)p(2-d)) distance queries with high probability, and this is best possible. 2012 ACM Subject Classification Mathematics of computing -> Random graphs; Mathematics of computing -> Graph algorithms

We show that for every ϵ>0 there exists a sufficiently large d_0∈ℕ such that for every d≥ d_0, whp the random d-regular graph G(n,d) contains a T-factor for every tree T on at most (1-ϵ)d/log d vertices. This is best possible since, for large enough integer d, whp G(n,d) does not contain a (1+ϵ)d/log d-star factor.