arXiv subjects

Michael Anastos

Publications and source records attributed to Michael Anastos.

At least 19 recordsLinked to original sources

arXiv2025

Diameter and mixing time of the giant component in the percolated hypercube

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 $\Theta(d)$, and the typical mixing time of the lazy random walk on $L_1$ is of order $\Theta(d^2)$. This resolves long-standing open problems of Bollob\'as, Kohayakawa and {\L}uczak 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 $L_1$.

math.PR↗

arXiv2025

Optimally building spanning graphs in semirandom graph processes

The semirandom graph process constructs a graph $G$ in a series of rounds, starting with the empty graph on $n$ vertices. In each round, a player is offered a vertex $v$ chosen uniformly at random, and chooses an edge on $v$ to add to $G$. The player's aim is to make $G$ satisfy some property as quickly as possible. Our interest is in the property that $G$ contain a given $n$-vertex graph $H$ with maximum degree $\Delta$. In 2021, Ben-Eliezer, Gishboliner, Hefetz and Krivelevich showed that there is a semirandom strategy that achieves this, with probability tending to 1 as $n$ tends to infinity, in $(1 + o_\Delta(1)) \frac{3 \Delta n}{2}$ rounds, where $o_\Delta(1)$ is a function that tends to $0$ as $\Delta$ tends to infinity. We improve this to $(1 + o_\Delta(1)) \frac{\Delta n}{2}$, which can be seen to be asymptotically optimal in $\Delta$. We show the same result for a variant of the semirandom graph process, namely the semirandom tree process introduced by Burova and Lichev, where in each round the player is offered the edge set of a uniformly chosen tree on $n$ vertices, and chooses one edge to keep.

math.CO↗

arXiv2025

Cycle lengths in the percolated hypercube

Let $Q^d_p$ be the random subgraph of the $d$-dimensional binary hypercube obtained after edge-percolation with probability $p$. It was shown recently by the authors that, for every $\varepsilon > 0$, there is some $c = c(\varepsilon)>0$ such that, if $pd\ge c$, then typically $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. We strengthen this result to show that, under the same assumptions, typically $Q^d_p$ contains cycles of all even lengths between $4$ and $(1-\varepsilon)2^d$.

math.CO↗

arXiv2025

A short proof of a central limit theorem for the order of the giant component and $k$-core

In this note we outline a new and simple approach to proving central limit theorems for various 'global' graph parameters which have robust 'local' approximations, using the Efron--Stein inequality, which relies on a combinatorial analysis of the stability of these approximations under resampling an edge. As an application, we give short proofs of a central limit theorem for the order of the giant component and of the $k$-core for sparse random graphs.

math.CO↗

arXiv2025

Spanning trees of bounded degree in random geometric graphs

We determine the sharp threshold for the containment of all $n$-vertex trees of bounded degree in random geometric graphs with $n$ vertices. This provides a geometric counterpart of Montgomery's threshold result for binomial random graphs, and confirms a conjecture of Espuny D\'iaz, Lichev, Mitsche, and Wesolek. Our proof is algorithmic and adapts to other families of graphs, in particular graphs with bounded genus or tree-width.

math.CO↗

arXiv2025

Nearly spanning cycle in the percolated hypercube

Let $Q^d$ be the $d$-dimensional binary hypercube. We form a random subgraph $Q^d_p\subseteq Q^d$ by retaining each edge of $Q^d$ independently with probability $p$. We show that, for every constant $\varepsilon>0$, there exists a constant $C=C(\varepsilon)>0$ such that, if $p\ge C/d$, then with high probability $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. This confirms a long-standing folklore conjecture, stated in particular by Condon, Espuny D\'iaz, Gir\~ao, K\"uhn, and Osthus [Hamiltonicity of random subgraphs of the hypercube, Mem. Amer. Math. Soc. 305 (2024), No. 1534].

math.CO↗

arXiv2025

The law of the circumference of sparse binomial random graphs

There has been much interest in the distribution of the circumference, the length of the longest cycle, of a random graph $G(n,p)$ in the sparse regime, when $p = \Theta\left(\frac{1}{n}\right)$. Recently, the first author and Frieze established a scaling limit for the circumference in this regime, along the way establishing an alternative 'structural' approximation for this parameter. In this paper, we give a central limit theorem for the circumference in this regime using a novel argument based on the Efron-Stein inequality, which relies on a combinatorial analysis of the effect of resampling edges on this approximation.

math.CO↗

arXiv2024

A note on finding large transversals efficiently

In an $n \times n$ array filled with symbols, a transversal is a collection of entries with distinct rows, columns and symbols. In this note we show that if no symbol appears more than $\beta n$ times, the array contains a transversal of size $(1-\beta/4-o(1))n$. In particular, if the array is filled with $n$ symbols, each appearing $n$ times (an equi-$n$ square), we get transversals of size $(3/4-o(1))n$. Moreover, our proof gives a deterministic algorithm with polynomial running time, that finds these transversals.

math.CO↗

arXiv2024

Smoothed analysis for graph isomorphism

There is no known polynomial-time algorithm for graph isomorphism testing, but elementary combinatorial "refinement" algorithms seem to be very efficient in practice. Some philosophical justification is provided by a classical theorem of Babai, Erd\H{o}s and Selkow: an extremely simple polynomial-time combinatorial algorithm (variously known as "na\"ive refinement", "na\"ive vertex classification", "colour refinement" or the "1-dimensional Weisfeiler-Leman algorithm") yields a so-called canonical labelling scheme for "almost all graphs". More precisely, for a typical outcome of a random graph $G(n,1/2)$, this simple combinatorial algorithm assigns labels to vertices in a way that easily permits isomorphism-testing against any other graph. We improve the Babai-Erd\H{o}s-Selkow theorem in two directions. First, we consider randomly perturbed graphs, in accordance with the smoothed analysis philosophy of Spielman and Teng: for any graph $G$, na\"ive refinement becomes effective after a tiny random perturbation to $G$ (specifically, the addition and removal of $O(n\log n)$ random edges). Actually, with a twist on na\"ive refinement, we show that $O(n)$ random additions and removals suffice. These results significantly improve on previous work of Gaudio-R\'acz-Sridhar, and are in certain senses best-possible. Second, we complete a long line of research on canonical labelling of random graphs: for any $p$ (possibly depending on $n$), we prove that a random graph $G(n,p)$ can typically be canonically labelled in polynomial time. This is most interesting in the extremely sparse regime where $p$ has order of magnitude $c/n$; denser regimes were previously handled by Bollob\'as, Czajka-Pandurangan, and Linial-Mosheiff. Our proof also provides a description of the automorphism group of a typical outcome of $G(n,p_n)$ (slightly correcting a prediction of Linial-Mosheiff).

math.CO↗

arXiv2024

On the chromatic number of powers of subdivisions of graphs

For a given graph $G=(V,E)$, we define its \emph{$n$th subdivision} as the graph obtained from $G$ by replacing every edge by a path of length $n$. We also define the \emph{$m$th power} of $G$ as the graph on vertex set $V$ where we connect every pair of vertices at distance at most $m$ in $G$. In this paper, we study the chromatic number of powers of subdivisions of graphs and resolve the case $m=n$ asymptotically. In particular, our result confirms a conjecture of Mozafari-Nia and Iradmusa in the case $m=n=3$ in a strong sense.

math.CO↗

arXiv2023

Climbing up a random subgraph of the hypercube

Let $Q^d$ be the $d$-dimensional binary hypercube. We say that $P=\{v_1,\ldots, v_k\}$ is an increasing path of length $k-1$ in $Q^d$, if for every $i\in [k-1]$ the edge $v_iv_{i+1}$ is obtained by switching some zero coordinate in $v_i$ to a one coordinate in $v_{i+1}$. Form a random subgraph $Q^d_p$ by retaining each edge in $E(Q^d)$ independently with probability $p$. We show that there is a phase transition with respect to the length of a longest increasing path around $p=\frac{e}{d}$. Let $\alpha$ be a constant and let $p=\frac{\alpha}{d}$. When $\alpha e$, whp there is a path of length $d-2$ in $Q^d_p$, and in fact, whether it is of length $d-2, d-1$, or $d$ depends on whether the all-zero and all-one vertices percolate or not.

math.CO↗

arXiv2023

Robust Hamiltonicity in families of Dirac graphs

A graph is called Dirac if its minimum degree is at least half of the number of vertices in it. Joos and Kim showed that every collection $\mathbb{G}=\{G_1,\ldots,G_n\}$ of Dirac graphs on the same vertex set $V$ of size $n$ contains a Hamilton cycle transversal, i.e., a Hamilton cycle $H$ on $V$ with a bijection $\phi:E(H)\rightarrow [n]$ such that $e\in G_{\phi(e)}$ for every $e\in E(H)$. In this paper, we determine up to a multiplicative constant, the threshold for the existence of a Hamilton cycle transversal in a collection of random subgraphs of Dirac graphs in various settings. Our proofs rely on constructing a spread measure on the set of Hamilton cycle transversals of a family of Dirac graphs. As a corollary, we obtain that every collection of $n$ Dirac graphs on $n$ vertices contains at least $(cn)^{2n}$ different Hamilton cycle transversals $(H,\phi)$ for some absolute constant $c>0$. This is optimal up to the constant $c$. Finally, we show that if $n$ is sufficiently large, then every such collection spans $n/2$ pairwise edge-disjoint Hamilton cycle transversals, and this is best possible. These statements generalize classical counting results of Hamilton cycles in a single Dirac graph.

math.CO↗

arXiv2023

Extremal, enumerative and probabilistic results on ordered hypergraph matchings

An ordered $r$-matching is an $r$-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of $r$-dimensional orders. The theory of ordered 2-matchings is well-developed and has connections and applications to extremal and enumerative combinatorics, probability, and geometry. On the other hand, in the case $r \ge 3$ much less is known, largely due to a lack of powerful bijective tools. Recently, Dudek, Grytczuk and Ruci\'nski made some first steps towards a general theory of ordered $r$-matchings, and in this paper we substantially improve several of their results and introduce some new directions of study. Many intriguing open questions remain.

math.CO↗

arXiv2023

Partitioning problems via random processes

There are a number of well-known problems and conjectures about partitioning graphs to satisfy local constraints. For example, the majority colouring conjecture of Kreutzer, Oum, Seymour, van der Zypen and Wood states that every directed graph has a 3-colouring such that for every vertex $v$, at most half of the out-neighbours of $v$ have the same colour as $v$. As another example, the internal partition conjecture, due to DeVos and to Ban and Linial, states that for every $d$, all but finitely many $d$-regular graphs have a partition into two nonempty parts such that for every vertex $v$, at least half of the neighbours of $v$ lie in the same part as $v$. We prove several results in this spirit: in particular, two of our results are that the majority colouring conjecture holds for Erd\H{o}s-R\'enyi random directed graphs (of any density), and that the internal partition conjecture holds if we permit a tiny number of "exceptional vertices". Our proofs involve a variety of techniques, including several different methods to analyse random recolouring processes. One highlight is a "personality-changing" scheme: we "forget" certain information based on the state of a Markov chain, giving us more independence to work with.

math.CO↗

arXiv2023

Splitting matchings and the Ryser-Brualdi-Stein conjecture for multisets

We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3\in \mathbb{N}$ with $a_1+a_2+a_3\leq n-2$, we show there exists a matching $M$ of $G$ with $|M\cap M_i|=a_i$ for each $i\in \{1,2,3\}$. The bound $n-2$ in the theorem is best possible in general. We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities.

math.CO↗

arXiv2023

The completion numbers of Hamiltonicity and pancyclicity in random graphs

Let $\mu(G)$ denote the minimum number of edges whose addition to $G$ results in a Hamiltonian graph, and let $\hat{\mu}(G)$ denote the minimum number of edges whose addition to $G$ results in a pancyclic graph. We study the distributions of $\mu (G),\hat{\mu}(G)$ in the context of binomial random graphs. Letting $d=d(n) := n\cdot p$, we prove that there exists a function $f:\mathbb{R}^+\to [0,1]$ of order $f(d) = \frac{1}{2}de^{-d}+e^{-d}+O(d^6e^{-3d})$ such that, if $G\sim G(n,p)$ with $20 \le d(n) \le 0.4 \log n$, then with high probability $\mu (G)= (1+o(1))\cdot f(d)\cdot n$. Let $n_i(G)$ denote the number of degree $i$ vertices in $G$. A trivial lower bound on $\mu(G)$ is given by the expression $n_0(G) + \lceil \frac{1}{2}n_1(G) \rceil$. In the denser regime of random graphs, we show that if $np-\frac{1}{3}\log n - 2\log \log n \to \infty$ and $G\sim G(n,p)$ then, with high probability, $\mu (G) = n_0(G) + \lceil \frac{1}{2}n_1(G) \rceil$. For completion to pancyclicity, we show that if $G\sim G(n,p)$ and $np\ge 20$ then, with high probability, $\hat{\mu} (G)=\mu (G)$. Finally, we present a polynomial time algorithm such that, if $G\sim G(n,p)$ and $np\ge 20$, then, with high probability, the algorithm returns a set of edges of size $\mu (G)$ whose addition to $G$ results in a pancyclic (and therefore also Hamiltonian) graph.

math.CO↗

arXiv2023

A note on long cycles in sparse random graphs

Let $L_{c,n}$ denote the size of the longest cycle in $G(n,{c}/{n})$, $c>1$ constant. We show that there exists a continuous function $f(c)$ such that $ L_{c,n}/n \to f(c)$ a.s. for $c\geq 20$, thus extending a result of the author and Frieze to smaller values of $c$. Thereafter, for $c\geq 20$, we determine the limit of the probability that $G(n,c/n)$ contains cycles of every length between the length of its shortest and its longest cycles as $n\to \infty$.

math.CO↗

arXiv2022

A fast algorithm on average for solving the Hamilton Cycle problem

We present CertifyHAM, an algorithm which takes as input a graph G and either finds a Hamilton cycle of G or it outputs that such a cycle does not exists. If G=G(n, p) and p >2000/n then the expected running time of CertifyHAM is O(n/p). This improves upon previous results due to Gurevich and Shelah, Thomason and Alon and Krivelevich.

math.CO↗