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Dan Hefetz

Publications and source records attributed to Dan Hefetz.

At least 19 recordsLinked to original sources
arXiv2026

On graphs whose cycle space is spanned by their Hamilton cycles

The cycle space of a graph $G$, denoted $\mathcal{C}(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $\mathcal{C}_n(G)$ denotes the subspace of $\mathcal{C}(G)$, spanned by the incidence vectors of Hamilton cycles of $G$. We consider several known sufficient conditions for Hamiltonicity and show that an appropriate and fairly mild strengthening of each such condition in fact ensures the stronger property $\mathcal{C}_n(G) = \mathcal{C}(G)$. In particular, we consider the classical Chv\'atal-Erd\H{o}s criterion and prove that (under various additional restrictions) if $n$ is odd and $\kappa(G) \geq c \alpha(G)$, where $c$ is a sufficiently large absolute constant, then $\mathcal{C}_n(G) = \mathcal{C}(G)$. Moreover, considering the McDiarmid-Yolov criterion we prove that if $n$ is odd and $\delta(G) \geq \max \left\{2 \tilde{\alpha}(G) + 9, \tilde{\alpha}(G) + 18 \right\}$, where $\tilde{\alpha}(G)$ is the so-called bipartite independence number of $G$, then $\mathcal{C}_n(G) = \mathcal{C}(G)$. We also prove that if $n$ is odd and $G$ admits $16 \alpha(G) + 12$ pairwise disjoint connected dominating sets, $\mathcal{C}_n(G) = \mathcal{C}(G)$. Finally, we consider an effective Chv\'atal-Erd\H{o}s type criterion for bipartite graphs and prove that if $G$ is a balanced bipartite graph on $2n$ vertices, satisfying $\alpha_{\emph{BIP}}(G) \leq 2 \delta(G) - 24$, then $\mathcal{C}_{2n}(G) = \mathcal{C}(G)$.

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arXiv2025

The Hamilton cycle space of random regular graphs and randomly perturbed graphs

The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ is the subspace of $C(G)$, spanned by the incidence vectors of Hamilton cycles of $G$. We prove that asymptotically almost surely $C_n(G_{n,d}) = C(G_{n,d})$ holds whenever $n$ is odd and $d$ is a sufficiently large (even) integer. This extends (though with a weaker bound on $d$) the well-known result asserting that $G_{n,d}$ is asymptotically almost surely Hamiltonian for every $d \geq 3$ (but not for $d < 3$). Since $n$ being odd mandates that $d$ be even, somewhat limiting the generality of our result, we also prove that if $n$ is even and $d$ is any sufficiently large integer, then asymptotically almost surely $C_{n-1}(G_{n,d}) = C(G_{n,d})$. An influential result of Bohman, Frieze, and Martin asserts that if $H$ is an $n$-vertex graph with minimum degree at least $\delta n$ for some constant $\delta > 0$, and $G \sim \mathbb{G}(n, C/n)$, where $C := C(\delta)$ is a sufficiently large constant, then $H \cup G$ is asymptotically almost surely Hamiltonian. We strengthen this result by proving that the same assumptions on $H$ and $G$ ensure that $C_n(H \cup G) = C(H \cup G)$ holds asymptotically almost surely.

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arXiv2025

The Hamilton cycle space of random graphs

The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ denotes the subspace of $C(G)$, spanned by the incidence vectors of Hamilton cycles of $G$. A classical result in the theory of random graphs asserts that for $G \sim \mathbb{G}(n,p)$, asymptotically almost surely the necessary condition $\delta(G) \geq 2$ is also sufficient to ensure Hamiltonicity. Resolving a problem of Christoph, Nenadov, and Petrova, we augment this result by proving that for $G \sim \mathbb{G}(n,p)$, with $n$ being odd, asymptotically almost surely the condition $\delta(G) \geq 3$ (observed to be necessary by Heinig) is also sufficient for ensuring $C_n(G) = C(G)$. That is, not only does $G$ typically have a Hamilton cycle, but its Hamilton cycles are typically rich enough to span its cycle space.

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arXiv2025

Smoothed analysis in compressed sensing

Arbitrary matrices $M \in \mathbb{R}^{m \times n}$, randomly perturbed in an additive manner using a random matrix $R \in \mathbb{R}^{m \times n}$, are shown to asymptotically almost surely satisfy the so-called {\sl robust null space property}. Whilst insisting on an asymptotically optimal order of magnitude for $m$ required to attain {\sl unique reconstruction} via $\ell_1$-minimisation algorithms, our results track the level of arbitrariness allowed for the fixed seed matrix $M$ as well as the degree of distributional irregularity allowed for the entries of the perturbing matrix $R$. Starting with sub-gaussian entries for $R$, our results culminate with these allowed to have substantially heavier tails than sub-exponential ones. Throughout this trajectory, two measures control the arbitrariness allowed for $M$; the first is $\|M\|_\infty$ and the second is a localised notion of the Frobenius norm of $M$ (which depends on the sparsity of the signal being reconstructed). A key tool driving our proofs is {\sl Mendelson's small-ball method} ({\em Learning without concentration}, J. ACM, Vol. $62$, $2015$).

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arXiv2024

Colouring graphs from random lists

Given positive integers $k \leq m$ and a graph $G$, a family of lists $L = \{L(v) : v \in V(G)\}$ is said to be a random $(k,m)$-list-assignment if for every $v \in V(G)$ the list $L(v)$ is a subset of $\{1, \ldots, m\}$ of size $k$, chosen uniformly at random and independently of the choices of all other vertices. An $n$-vertex graph $G$ is said to be a.a.s. $(k,m)$-colourable if $\lim_{n \to \infty} \mathbb{P}(G \textrm{ is } L-colourable) = 1$, where $L$ is a random $(k,m)$-list-assignment. We prove that if $m \gg n^{1/k^2} \Delta^{1/k}$ and $m \geq 3 k^2 \Delta$, where $\Delta$ is the maximum degree of $G$ and $k \geq 3$ is an integer, then $G$ is a.a.s. $(k,m)$-colourable. This is not far from being best possible, forms a continuation of the so-called palette sparsification results, and proves in a strong sense a conjecture of Casselgren. Additionally, we consider this problem under the additional assumption that $G$ is $H$-free for some graph $H$. For various graphs $H$, we estimate the smallest $m$ for which an $H$-free $n$-vertex graph $G$ is a.a.s. $(k,m)$-colourable. This extends and improves several results of Casselgren.

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arXiv2023

Ramsey properties of randomly perturbed hypergraphs

We study Ramsey properties of randomly perturbed $3$-uniform hypergraphs. For~$t\geq 2$, write $\tilde K^{(3)}_t$ to denote the $3$-uniform {\it expanded} clique hypergraph obtained from the complete graph $K_t$ by expanding each of the edges of the latter with a new additional vertex. For an even integer $t\geq 4$, let~$M$ denote the asymmetric maximal density of the pair $(\tilde K^{(3)}_t,\tilde K^{(3)}_{t/2})$. We prove that adding a set~$F$ of random hyperedges satisfying $|F|\gg n^{3-1/M}$ to a given $n$-vertex $3$-uniform hypergraph~$H$ with non-vanishing edge density asymptotically almost surely results in a perturbed hypergraph enjoying the Ramsey property for $\tilde K^{(3)}_t$ and two colours. We conjecture that this result is asymptotically best possible with respect to the size of $F$ whenever $t\geq 6$ is even. The key tools of our proof are a new variant of the hypergraph regularity lemma accompanied with a \emph{tuple lemma} providing appropriate control over joint link graphs. Our variant combines the so called strong and the weak hypergraph regularity lemmata.

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arXiv2023

Smoothed Analysis of the Koml\'os Conjecture: Rademacher Noise

The {\em discrepancy} of a matrix $M \in \mathbb{R}^{d \times n}$ is given by $\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n} \|M\boldsymbol{x}\|_\infty$. An outstanding conjecture, attributed to Koml\'os, stipulates that $\mathrm{DISC}(M) = O(1)$, whenever $M$ is a Koml\'os matrix, that is, whenever every column of $M$ lies within the unit sphere. Our main result asserts that $\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2})$ holds asymptotically almost surely, whenever $M \in \mathbb{R}^{d \times n}$ is Koml\'os, $R \in \mathbb{R}^{d \times n}$ is a Rademacher random matrix, $d = \omega(1)$, and $n = \omega(d \log d)$. The factor $d^{-1/2}$ normalising $R$ is essentially best possible and the dependency between $n$ and $d$ is asymptotically best possible. Our main source of inspiration is a result by Bansal, Jiang, Meka, Singla, and Sinha (ICALP 2022). They obtained an assertion similar to the one above in the case that the smoothing matrix is Gaussian. They asked whether their result can be attained with the optimal dependency $n = \omega(d \log d)$ in the case of Bernoulli random noise or any other types of discretely distributed noise; the latter types being more conducive for Smoothed Analysis in other discrepancy settings such as the Beck-Fiala problem. For Bernoulli noise, their method works if $n = \omega(d^2)$. In the case of Rademacher noise, we answer the question posed by Bansal, Jiang, Meka, Singla, and Sinha. Our proof builds upon their approach in a strong way and provides a discrete version of the latter. Breaking the $n = \omega(d^2)$ barrier and reaching the optimal dependency $n = \omega(d \log d)$ for Rademacher noise requires additional ideas expressed through a rather meticulous counting argument, incurred by the need to maintain a high level of precision all throughout the discretisation process.

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arXiv2022

Minors, connectivity, and diameter in randomly perturbed sparse graphs

Extremal properties of sparse graphs, randomly perturbed by the binomial random graph are considered. It is known that every $n$-vertex graph $G$ contains a complete minor of order $\Omega(n/\alpha(G))$. We prove that adding $\xi n$ random edges, where $\xi > 0$ is arbitrarily small yet fixed, to an $n$-vertex graph $G$ satisfying $\alpha(G) \leq \zeta(\xi)n$ asymptotically almost surely results in a graph containing a complete minor of order $\tilde \Omega \left( n/\sqrt{\alpha(G)}\right)$; this result is tight up to the implicit logarithmic terms. For complete topological minors, we prove that there exists a constant $C>0$ such that adding $C n$ random edges to a graph $G$ satisfying $\delta(G) = \omega(1)$, asymptotically almost surely results in a graph containing a complete topological minor of order $\tilde \Omega(\min\{\delta(G),\sqrt{n}\})$; this result is tight up to the implicit logarithmic terms. Finally, extending results of Bohman, Frieze, Krivelevich, and Martin for the dense case, we analyse the asymptotic behaviour of the vertex-connectivity and the diameter of randomly perturbed sparse graphs.

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arXiv2022

Cycle lengths in randomly perturbed graphs

Let $G$ be an $n$-vertex graph, where $\delta(G) \geq \delta n$ for some $\delta := \delta(n)$. A result of Bohman, Frieze and Martin from 2003 asserts that if $\alpha(G) = O \left(\delta^2 n \right)$, then perturbing $G$ via the addition of $\omega \left(\frac{\log(1/\delta)}{\delta^3} \right)$ random edges, asymptotically almost surely (a.a.s. hereafter) results in a Hamiltonian graph. This bound on the size of the random perturbation is only tight when $\delta$ is independent of $n$ and deteriorates as to become uninformative when $\delta = \Omega \left(n^{-1/3} \right)$. We prove several improvements and extensions of the aforementioned result. First, keeping the bound on $\alpha(G)$ as above and allowing for $\delta = \Omega(n^{-1/3})$, we determine the correct order of magnitude of the number of random edges whose addition to $G$ a.a.s. results in a pancyclic graph. Our second result ventures into significantly sparser graphs $G$; it delivers an almost tight bound on the size of the random perturbation required to ensure pancyclicity a.a.s., assuming $\delta(G) = \Omega \left((\alpha(G) \log n)^2 \right)$ and $\alpha(G) \delta(G) = O(n)$. Assuming the correctness of Chv\'atal's toughness conjecture, allows for the mitigation of the condition $\alpha(G) = O \left(\delta^2 n \right)$ imposed above, by requiring $\alpha(G) = O(\delta(G))$ instead; our third result determines, for a wide range of values of $\delta(G)$, the correct order of magnitude of the size of the random perturbation required to ensure the a.a.s. pancyclicity of $G$. For the emergence of nearly spanning cycles, our fourth result determines, under milder conditions, the correct order of magnitude of the size of the random perturbation required to ensure that a.a.s. $G$ contains such a cycle.

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arXiv2021

Rainbow trees in uniformly edge-coloured graphs

We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {\sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform colouring of $\mathbb{G}(n,\omega(1)/n)$, using a palette of size $n$, a.a.s. admits a rainbow copy of any given bounded-degree tree on at most $(1-\varepsilon)n$ vertices, where $\varepsilon > 0$ is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon, Krivelevich, and Sudakov pertaining to the embedding of bounded-degree almost-spanning prescribed trees in $\mathbb{G}(n,C/n)$, where $C > 0$ is independent of $n$. Given an $n$-vertex graph $G$ with minimum degree at least $\delta n$, where $\delta > 0$ is fixed, we use our aforementioned result in order to prove that a uniform colouring of the randomly perturbed graph $G \cup \mathbb{G}(n,\omega(1)/n)$, using $(1+\alpha)n$ colours, where $\alpha > 0$ is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded-degree {\sl spanning} tree. This can be viewed as a rainbow variant of a result by Krivelevich, Kwan, and Sudakov who proved that $G \cup \mathbb{G}(n,C/n)$, where $C > 0$ is independent of $n$, a.a.s. admits a copy of any given bounded-degree spanning tree. Finally, and with $G$ as above, we prove that a uniform colouring of $G \cup \mathbb{G}(n,\omega(n^{-2}))$ using $n-1$ colours a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

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arXiv2020

Semi-random process without replacement

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some pre-determined objective in an online randomized environment. We introduce and study a semi-random multigraph process, which forms a no-replacement variant of the process that was introduced by Ben-Eliezer, Hefetz, Kronenberg, Parczyk, Shikhelman and Stojakovi\'c (2020). The process starts with an empty graph on the vertex set $[n]$. For every positive integers $q$ and $1\leq r\leq n$, in the $((q-1)n+r)$th round of the process, the decision-maker, called \emph{Builder}, is offered the vertex $\pi_q(r)$, where $\pi_1, \pi_2, \ldots$ is a sequence of permutations in $S_n$, chosen independently and uniformly at random. Builder then chooses an additional vertex (according to a strategy of his choice) and connects it by an edge to $\pi_q(r)$. For several natural graph properties, such as $k$-connectivity, minimum degree at least $k$, and building a given spanning graph (labeled or unlabeled), we determine the typical number of rounds Builder needs in order to construct a graph having the desired property. Along the way we introduce and analyze an urn model which may also have independent interest.

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arXiv2020

Small rainbow cliques in randomly perturbed dense graphs

For two graphs $G$ and $H$, write $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ if $G$ has the property that every \emph{proper} colouring of its edges yields a \emph{rainbow} copy of $H$. We study the thresholds for such so-called \emph{anti-Ramsey} properties in randomly perturbed dense graphs, which are unions of the form $G \cup \mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least $d >0$, and $d$ is independent of $n$. In a companion article, we proved that the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_\ell$ is $n^{-1/m_2(K_{\left\lceil \ell/2 \right\rceil})}$, whenever $\ell \geq 9$. For smaller $\ell$, the thresholds behave more erratically, and for $4 \le \ell \le 7$ they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for \emph{large} cliques. In particular, we show that the thresholds for $\ell \in \{4, 5, 7\}$ are $n^{-5/4}$, $n^{-1}$, and $n^{-7/15}$, respectively. For $\ell \in \{6, 8\}$ we determine the threshold up to a $(1 + o(1))$-factor in the exponent: they are $n^{-(2/3 + o(1))}$ and $n^{-(2/5 + o(1))}$, respectively. For $\ell = 3$, the threshold is $n^{-2}$; this follows from a more general result about odd cycles in our companion paper.

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arXiv2020

Rainbow Hamilton cycles in randomly coloured randomly perturbed dense graphs

Given an $n$-vertex graph $G$ with minimum degree at least $d n$ for some fixed $d > 0$, the distribution $G \cup \mathbb{G}(n,p)$ over the supergraphs of $G$ is referred to as a (random) {\sl perturbation} of $G$. We consider the distribution of edge-coloured graphs arising from assigning each edge of the random perturbation $G \cup \mathbb{G}(n,p)$ a colour, chosen independently and uniformly at random from a set of colours of size $r := r(n)$. We prove that such edge-coloured graph distributions a.a.s. admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisfies $p := p(n) \geq C/n$, for some fixed $C > 0$, and $r = (1 + o(1))n$. The number of colours used is clearly asymptotically best possible. In particular, this improves upon a recent result of Anastos and Frieze (2019) in this regard. As an intermediate result, which may be of independent interest, we prove that randomly edge-coloured sparse pseudo-random graphs a.a.s. admit an almost spanning rainbow path.

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arXiv2019

Large rainbow cliques in randomly perturbed dense graphs

For two graphs $G$ and $H$, write $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ if $G$ has the property that every {\sl proper} colouring of its edges yields a {\sl rainbow} copy of $H$. We study the thresholds for such so-called {\sl anti-Ramsey} properties in randomly perturbed dense graphs, which are unions of the form $G \cup \mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least $d$, and $d$ is a constant that does not depend on $n$. Our results in this paper, combined with our results in a companion paper, determine the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_s$ for every $s$. In this paper, we show that for $s \geq 9$ the threshold is $n^{-1/m_2(K_{\left\lceil s/2 \right\rceil})}$; in fact, our $1$-statement is a supersaturation result. This turns out to (almost) be the threshold for $s=8$ as well, but for every $4 \leq s \leq 7$, the threshold is lower; see our companion paper for more details. In this paper, we also consider the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} C_{2\ell - 1}$, and show that the threshold for this property is $n^{-2}$ for every $\ell \geq 2$; in particular, it does not depend on the length of the cycle $C_{2\ell - 1}$. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.

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arXiv2019

On the local structure of oriented graphs -- a case study in flag algebras

Let $G$ be an $n$-vertex oriented graph. Let $t(G)$ (respectively $i(G)$) be the probability that a random set of $3$ vertices of $G$ spans a transitive triangle (respectively an independent set). We prove that $t(G) + i(G) \geq \frac{1}{9}-o_n(1)$. Our proof uses the method of flag algebras that we supplement with several steps that make it more easily comprehensible. We also prove a stability result and an exact result. Namely, we describe an extremal construction, prove that it is essentially unique, and prove that if $H$ is sufficiently far from that construction, then $t(H) + i(H)$ is significantly larger than $\frac{1}{9}$. We go to greater technical detail than is usually done in papers that rely on flag algebras. Our hope is that as a result this text can serve others as a useful introduction to this powerful and beautiful method.

math.CO↗
arXiv2019

Very fast construction of bounded-degree spanning graphs via the semi-random graph process

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some predetermined objective in an online randomized environment. They have algorithmic implications in various areas of computer science, as well as connections to biological processes involving decision making. In this paper, we consider a recently proposed semi-random graph process, described as follows: we start with an empty graph on $n$ vertices, and in each round, the decision-maker, called Builder, receives a uniformly random vertex $v$, and must immediately (in an online manner) choose another vertex $u$, adding the edge $\{u,v\}$ to the graph. Builder's end goal is to make the constructed graph satisfy some predetermined monotone graph property. We consider the property of containing a spanning graph $H$ as a subgraph. It was asked by N. Alon whether for any bounded-degree $H$, Builder can construct a copy of $H$ w.h.p. in $O(n)$ rounds. We answer this question positively in a strong sense, showing that any graph with maximum degree $\Delta$ can be constructed w.h.p. in $(3\Delta/2 + o(\Delta)) n$ rounds. This is tight (even for the offline case) up to a multiplicative factor of $3 + o_{\Delta}(1)$. Furthermore, for the special case where $H$ is a spanning forest of maximum degree $\Delta$, we show that $H$ can be constructed w.h.p. in $O(n \log \Delta)$ rounds. This is tight up to a multiplicative constant, even for the offline setting. Finally, we show a separation between adaptive and non-adaptive strategies, proving a lower bound of $\Omega(n\sqrt{\log n})$ on the number of rounds necessary to eliminate all isolated vertices w.h.p. using a non-adaptive strategy. This bound is tight, and in fact there are non-adaptive strategies for constructing a Hamilton cycle or a $K_r$-factor, which are successful w.h.p. within $O(n\sqrt{\log n})$ rounds.

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arXiv2018

Edge-statistics on large graphs

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$ vertices can have. Clearly, this number is $\binom{n}{k}$ for every $n$, $k$ and $\ell \in \left \{0, \binom{k}{2} \right\}$. We conjecture that for every $n$, $k$ and $0 < \ell < \binom{k}{2}$ this number is at most $\left(1/e + o_k(1) \right) \binom{n}{k}$. If true, this would be tight for $\ell \in \{1, k-1\}$. In support of our `Edge-statistics conjecture' we prove that the corresponding density is bounded away from $1$ by an absolute constant. Furthermore, for various ranges of the values of $\ell$ we establish stronger bounds. In particular, we prove that for `almost all' pairs $(k, \ell)$ only a polynomially small fraction of the $k$-subsets of $V(G)$ has exactly $\ell$ edges, and prove an upper bound of $(1/2 + o_k(1))\binom{n}{k}$ for $\ell = 1$. Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun's sieve, as well as graph-theoretic and combinatorial arguments such as Zykov's symmetrization, Sperner's theorem and various counting techniques.

math.CO↗
arXiv2018

Semi-random graph process

We introduce and study a novel semi-random multigraph process, described as follows. The process starts with an empty graph on $n$ vertices. In every round of the process, one vertex $v$ of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to $v$. For various natural monotone increasing graph properties $P$, we prove tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies $P$. Along the way, we show that the process is general enough to approximate (using suitable strategies) several well-studied random graph models.

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