arXiv subjects

Rajko Nenadov

Publications and source records attributed to Rajko Nenadov.

At least 55 records · Page 3Linked to original sources

arXiv2019

Complete minors in graphs without sparse cuts

We show that if $G$ is a graph on $n$ vertices, with all degrees comparable to some $d = d(n)$, and without a sparse cut, for a suitably chosen notion of sparseness, then it contains a complete minor of order \[ Ω\left( \sqrt{\frac{n d}{\log d}} \right). \] As a corollary we determine the order of a largest complete minor one can guarantee in $d$-regular graphs for which the second largest eigenvalue is bounded away from $d/2$, in $(d/n, o(d))$-jumbled graphs, and in random $d$-regular graphs, for almost all $d = d(n)$.

math.CO↗

arXiv2018

Resilience of Perfect Matchings and Hamiltonicity in Random Graph Processes

Let $\{G_i\}$ be the random graph process: starting with an empty graph $G_0$ with $n$ vertices, in every step $i \geq 1$ the graph $G_i$ is formed by taking an edge chosen uniformly at random among the non-existing ones and adding it to the graph $G_{i - 1}$. The classical `hitting-time' result of Ajtai, Komlós, and Szemerédi, and independently Bollobás, states that asymptotically almost surely the graph becomes Hamiltonian as soon as the minimum degree reaches $2$, that is if $δ(G_i) \ge 2$ then $G_i$ is Hamiltonian. We establish a resilience version of this result. In particular, we show that the random graph process almost surely creates a sequence of graphs such that for $m \geq (\tfrac{1}{6} + o(1))n\log n$ edges, the $2$-core of the graph $G_m$ remains Hamiltonian even after an adversary removes $(\tfrac{1}{2} - o(1))$-fraction of the edges incident to every vertex. A similar result is obtained for perfect matchings.

math.CO↗

arXiv2018

Monochromatic cycle covers in random graphs

A classic result of Erdős, Gyárfás and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In particular, the minimum number of such covering cycles does not depend on the size of $K_n$ but only on the number of colors. We initiate the study of this phenomena in the case where $K_n$ is replaced by the random graph $\mathcal G(n,p)$. Given a fixed integer $r$ and $p =p(n) \ge n^{-1/r + \varepsilon}$, we show that with high probability the random graph $G \sim \mathcal G(n,p)$ has the property that for every $r$-coloring of the edges of $G$, there is a collection of $f'(r) = O(r^8 \log r)$ monochromatic cycles covering all the vertices of $G$. Our bound on $p$ is close to optimal in the following sense: if $p\ll (\log n/n)^{1/r}$, then with high probability there are colorings of $G\sim\mathcal G(n,p)$ such that the number of monochromatic cycles needed to cover all vertices of $G$ grows with $n$.

math.CO↗

arXiv2018

On a Ramsey-Turán variant of the Hajnal-Szemerédi theorem

A seminal result of Hajnal and Szemerédi states that if a graph $G$ with $n$ vertices has minimum degree $δ(G) \ge (r-1)n/r$ for some integer $r \ge 2$, then $G$ contains a $K_r$-factor, assuming $r$ divides $n$. Extremal examples which show optimality of the bound on $δ(G)$ are very structured and, in particular, contain large independent sets. In analogy to the Ramsey-Turán theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results: $\bullet$ For any $r > \ell \ge 2$, if $G$ is a graph satisfying $δ(G) \ge (r - \ell)n/(r - \ell + 1) +Ω(n)$ and $α_\ell(G)=o(n)$, that is, a largest $K_\ell$-free induced subgraph has at most $o(n)$ vertices, then $G$ contains a $K_r$-factor. This is optimal for $\ell = r - 1$ and extends a result of Balogh, Molla, and Sharifzadeh who considered the case $r = 3$. $\bullet$ If a graph $G$ satisfies $δ(G) =Ω(n)$ and $α_r^*(G) =o(n)$, that is, every induced $K_r$-free $r$-partite subgraph of $G$ has at least one vertex class of size $o(n)$, then it contains a $K_r$-factor. A similar statement is proven for a general graph $H$.

math.CO↗

arXiv2018

Triangle-factors in pseudorandom graphs

We show that if the second eigenvalue $λ$ of a $d$-regular graph $G$ on $n \in 3 \mathbb{Z}$ vertices is at most $\varepsilon d^2/(n \log n)$, for a small constant $\varepsilon > 0$, then $G$ contains a triangle-factor. The bound on $λ$ is at most an $O(\log n)$ factor away from the best possible one: Krivelevich, Sudakov and Szabó, extending a construction of Alon, showed that for every function $d = d(n)$ such that $Ω(n^{2/3}) \le d \le n$ and infinitely many $n \in \mathbb{N}$ there exists a $d$-regular triangle-free graph $G$ with $Θ(n)$ vertices and $λ= Ω(d^2 / n)$.

math.CO↗

arXiv2017

Spanning universality in random graphs

A graph is said to be $\mathcal{H}(n, Δ)$-universal if it contains every graph on $n$ vertices with maximum degree at most $Δ$. Using a `matching-based' embedding technique introduced by Alon and Füredi, Dellamonica, Kohayakawa, Rödl and Ruciński showed that the random graph $G_{n,p}$ is asymptotically almost surely $\mathcal{H}(n, Δ)$-universal for $p = \tilde Ω(n^{-1/Δ})$ - a threshold for the property that every subset of $Δ$ vertices has a common neighbour. This bound has become a benchmark in the field and many subsequent results on embedding spanning structures of maximum degree $Δ$ in random graphs are proven only up to this threshold. We take a step towards overcoming limitations of former techniques by showing that $G_{n,p}$ is almost surely $\mathcal{H}(n, Δ)$-universal for $p = \tilde Ω(n^{- 1/(Δ-1/2)})$.

math.CO↗

arXiv2017

Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

We show that for every $k \in \mathbb{N}$ there exists $C > 0$ such that if $p^k \ge C \log^8 n / n$ then asymptotically almost surely the random graph $G_{n,p}$ contains the $k$\textsuperscript{th} power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kühn and Osthus. Moreover, our proof provides a randomized quasi-polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi-polynomial algorithm for finding a tight Hamilton cycle in the random $k$-uniform hypergraph $G_{n,p}^{(k)}$ for $p \ge C \log^8 n/ n$. The proofs are based on the absorbing method and follow the strategy of Kühn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of $p$. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.

math.CO↗

arXiv2016

On Komlós' tiling theorem in random graphs

Conlon, Gowers, Samotij, and Schacht showed that for a given graph $H$ and a constant $γ> 0$, there exists $C > 0$ such that if $p \ge Cn^{-1/m_2(H)}$ then asymptotically almost surely every spanning subgraph $G$ of the random graph $\mathcal{G}(n,p)$ with minimum degree at least $δ(G) \ge (1 - 1/χ_{\mathrm{cr}}(H) + γ)np$ contains an $H$-packing that covers all but at most $γn$ vertices. Here, $χ_{\mathrm{cr}}(H)$ denotes the critical chromatic threshold, a parameter introduced by Komlós. We show that this theorem can be bootstraped to obtain an $H$-packing covering all but at most $γ(C/p)^{m_2(H)}$ vertices, which is strictly smaller when $p > C n^{-1/m_2(H)}$. In the case where $H = K_3$ this answers the question of Balogh, Lee, and Samotij. Furthermore, we give an upper bound on the size of an $H$-packing for certain ranges of $p$.

math.CO↗

arXiv2016

Star-factors in graphs of high degree

We prove that every graph with sufficiently large minimum degree $d$ contains a spanning forest in which every component is a star of size at least $\sqrt{d} - \tilde O(d^{1/4})$. This improves the result of Alon and Wormald and is optimal up to the lower order term.

math.CO↗

arXiv2016

Symmetric and asymmetric Ramsey properties in random hypergraphs

A celebrated result of Rödl and Ruciński states that for every graph $F$, which is not a forest of stars and paths of length $3$, and fixed number of colours $r\ge 2$ there exist positive constants $c, C$ such that for $p \leq cn^{-1/m_2(F)}$ the probability that every colouring of the edges of the random graph $G(n,p)$ contains a monochromatic copy of $F$ is $o(1)$ (the "0-statement"), while for $p \geq Cn^{-1/m_2(F)}$ it is $1-o(1)$ (the "1-statement"). Here $m_2(F)$ denotes the $2$-density of $F$. On the other hand, the case where $F$ is a forest of stars has a coarse threshold which is determined by the appearance of a certain small subgraph in $G(n, p)$. Recently, the natural extension of the 1-statement of this theorem to $k$-uniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, Rödl and Schacht. In particular, they showed an upper bound of order $n^{-1/m_k(F)}$ for the $1$-statement, where $m_k(F)$ denotes the $k$-density of $F$. Similarly as in the graph case, it is known that the threshold for star-like hypergraphs is given by the appearance of small subgraphs. In this paper we show that another type of thresholds exists if $k \ge 4:$ there are $k$-uniform hypergraphs for which the threshold is determined by the asymmetric Ramsey problem in which a different hypergraph has to be avoided in each colour-class. Along the way we obtain a general bound on the $1$-statement for asymmetric Ramsey properties in random hypergraphs. This extends the work of Kohayakawa and Kreuter, and of Kohayakawa, Schacht and Spöhel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.

math.CO↗

arXiv2016

Optimal induced universal graphs for bounded-degree graphs

We show that for any constant $Δ\ge 2$, there exists a graph $G$ with $O(n^{Δ/ 2})$ vertices which contains every $n$-vertex graph with maximum degree $Δ$ as an induced subgraph. For odd $Δ$ this significantly improves the best-known earlier bound of Esperet et al. and is optimal up to a constant factor, as it is known that any such graph must have at least $Ω(n^{Δ/2})$ vertices. Our proof builds on the approach of Alon and Capalbo (SODA 2008) together with several additional ingredients. The construction of $G$ is explicit and is based on an appropriately defined composition of high-girth expander graphs. The proof also provides an efficient deterministic procedure for finding, for any given input graph $H$ on $n$ vertices with maximum degree at most $Δ$, an induced subgraph of $G$ isomorphic to $H$.

math.CO↗

arXiv2016

Unique reconstruction threshold for random jigsaw puzzles

A random jigsaw puzzle is constructed by arranging $n^2$ square pieces into an $n \times n$ grid and assigning to each edge of a piece one of $q$ available colours uniformly at random, with the restriction that touching edges receive the same colour. We show that if $q = o(n)$ then with high probability such a puzzle does not have a unique solution, while if $q \ge n^{1 + \varepsilon}$ for any constant $\varepsilon > 0$ then the solution is unique. This solves a conjecture of Mossel and Ross (Shotgun assembly of labeled graphs, arXiv:1504.07682).

cs.DM↗

arXiv2016

Almost-spanning universality in random graphs

A graph $G$ is said to be $\mathcal H(n,Δ)$-universal if it contains every graph on $n$ vertices with maximum degree at most $Δ$. It is known that for any $\varepsilon > 0$ and any natural number $Δ$ there exists $c > 0$ such that the random graph $G(n,p)$ is asymptotically almost surely $\mathcal H((1-\varepsilon)n,Δ)$-universal for $p \geq c (\log n/n)^{1/Δ}$. Bypassing this natural boundary, we show that for $Δ\geq 3$ the same conclusion holds when $p = ω\left(n^{-\frac{1}{Δ-1}}\log^5 n\right)$.

math.CO↗

arXiv2014

Robust hamiltonicity of random directed graphs

In his seminal paper from 1952 Dirac showed that the complete graph on $n\geq 3$ vertices remains Hamiltonian even if we allow an adversary to remove $\lfloor n/2\rfloor$ edges touching each vertex. In 1960 Ghouila-Houri obtained an analogue statement for digraphs by showing that every directed graph on $n\geq 3$ vertices with minimum in- and out-degree at least $n/2$ contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle. A natural way to generalize such results to arbitrary graphs (digraphs) is using the notion of \emph{local resilience}. The local resilience of a graph (digraph) $G$ with respect to a property $\mathcal{P}$ is the maximum number $r$ such that $G$ has the property $\mathcal{P}$ even if we allow an adversary to remove an $r$-fraction of (in- and out-going) edges touching each vertex. The theorems of Dirac and Ghouila-Houri state that the local resilience of the complete graph and digraph with respect to Hamiltonicity is $1/2$. Recently, this statements have been generalized to random settings. Lee and Sudakov (2012) proved that the local resilience of a random graph with edge probability $p=ω(\log n /n)$ with respect to Hamiltonicity is $1/2\pm o(1)$. For random directed graphs, Hefetz, Steger and Sudakov (2014+) proved an analogue statement, but only for edge probability $p=ω(\log n/\sqrt{n})$. In this paper we significantly improve their result to $p=ω(\log^8 n/ n)$, which is optimal up to the polylogarithmic factor.

math.CO↗

arXiv2014

Universality of random graphs and rainbow embedding

In this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of $G(n,p)$. Let the \emph{maximum density} of a graph $H$ be the maximum average degree of all the subgraphs of $H$. First, we show that for $p=ω(Δ^{12} n^{-1/2d}\log^3n)$, a graph $G\sim G(n,p)$ w.h.p.\ contains copies of all spanning graphs $H$ with maximum degree at most $Δ$ and maximum density at most $d$. For $d<Δ/2$, this improves a result of Dellamonica, Kohayakawa, Rödl and Rucińcki. Next, we show that if we additionally restrict the spanning graphs to have girth at least 7 then the random graph contains w.h.p.\ all such graphs for $p=ω(Δ^{12} n^{-1/d}\log^3n)$. In particular, if $p=ω(Δ^{12} n^{-1/2}\log^3 n)$, the random graph therefore contains w.h.p.\ every spanning tree with maximum degree bounded by $Δ$. This improves a result of Johannsen, Krivelevich and Samotij. Finally, in the same spirit, we show that for any spanning graph $H$ with constant maximum degree, and for suitable $p$, if we randomly color the edges of a graph $G\sim G(n,p)$ with $(1 + o(1))|E(H)|$ colors, then w.h.p.\ there exists a \emph{rainbow} copy of $H$ in $G$ (that is, a copy of $H$ with all edges colored with distinct colors).

math.CO↗

arXiv2014

An algorithmic framework for obtaining lower bounds for random Ramsey problems

In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that finds the desired colouring with high probability. Our framework allows to reduce the probabilistic problem of whether the Ramsey property at hand holds for random (hyper)graphs with edge probability $p$ to a deterministic question of whether there exists a finite graph that forms an obstruction. In the second part of the paper we apply this framework to address and solve various open problems. In particular, we extend the result of Bohman, Frieze, Pikhurko and Smyth (2010) for bounded anti-Ramsey problems in random graphs to the case of $2$ colors and to hypergraph cliques. As a corollary, this proves a matching lower bound for the result of Friedgut, Rödl and Schacht (2010) and, independently, Conlon and Gowers (2014+) for the classical Ramsey problem for hypergraphs in the case of cliques. Finally, we provide matching lower bounds for a proper-colouring version of anti-Ramsey problems introduced by Kohayakawa, Konstadinidis and Mota~(2014) in the case of cliques and cycles.

math.CO↗

arXiv2014

On the number of graphs without large cliques

In 1976 Erdos, Kleitman and Rothschild determined the number of graphs without a clique of size $\ell$. In this note we extend their result to the case of forbidden cliques of increasing size. More precisely we prove that for $\ell_n \le \frac12(\log n)^{1/4}$ there are $$2^{(1-1/(\ell_n-1))n^2/2+o(n^2/\ell_n)}$$ $K_{\ell_n}$-free graphs of order $n$. Our proof is based on the recent hypergraph container theorems of Saxton, Thomason and Balogh, Morris, Samotij, in combination with a theorem of Lovasz and Simonovits.

math.CO↗

arXiv2014

On the threshold for the Maker-Breaker $H$-game

We study the Maker-Breaker $H$-game played on the edge set of the random graph $G_{n,p}$. In this game two players, Maker and Breaker, alternately claim unclaimed edges of $G_{n,p}$, until all the edges are claimed. Maker wins if he claims all the edges of a copy of a fixed graph $H$; Breaker wins otherwise. In this paper we show that, with the exception of trees and triangles, the threshold for an $H$-game is given by the threshold of the corresponding Ramsey property of $G_{n,p}$ with respect to the graph $H$.

math.CO↗