arXiv subjects

József Balogh

Publications and source records attributed to József Balogh.

At least 91 records · Page 5Linked to original sources

arXiv2020

A discrepancy version of the Hajnal-Szemerédi theorem

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of the clique $K_r$ in $G$ covering every vertex of $G$. The famous Hajnal--Szemerédi theorem determines the minimum degree threshold for forcing a perfect $K_r$-tiling in a graph $G$. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph $G$ labels from $\{-1,1\}$, and one seeks substructures $F$ of $G$ that have `high' discrepancy (i.e. the sum of the labels of the edges in $F$ is far from $0$). In this paper we determine the minimum degree threshold for a graph to contain a perfect $K_r$-tiling of high discrepancy.

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arXiv2020

On the discrepancies of graphs

In the literature, the notion of discrepancy is used in several contexts, even in the theory of graphs. Here, for a graph $G$, $\{-1, 1\}$ labels are assigned to the edges, and we consider a family $\mathcal{S}_G$ of (spanning) subgraphs of certain types, among others spanning trees, Hamiltonian cycles. As usual, we seek for bounds on the sum of the labels that hold for all elements of $\mathcal{S}_G$, for every labeling.

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arXiv2020

The domination number of the graph defined by two levels of the $n$-cube, II

Consider all $k$-element subsets and $\ell$-element subsets $(k>\ell )$ of an $n$-element set as vertices of a bipartite graph. Two vertices are adjacent if the corresponding $\ell$-element set is a subset of the corresponding $k$-element set. Let $G_{k,\ell}$ denote this graph. The domination number of $G_{k,1}$ was exactly determined by Badakhshian, Katona and Tuza. A conjecture was also stated there on the asymptotic value ($n$ tending to infinity) of the domination number of $G_{k,2}$. Here we prove the conjecture, determining the asymptotic value of the domination number $γ(G_{k,2})={k+3\over 2(k-1)(k+1)}n^2+o(n^2)$.

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arXiv2020

An analogue of the Erdős-Gallai theorem for random graphs

Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erdős-Gallai Theorem in random graphs. In particular, we determine, up to a constant factor, the maximum number of edges in a $P_n$-free subgraph of $G(N,p)$, practically for all values of $N,n$ and $p$. Our work is also motivated by the recent progress on the size-Ramsey number of paths.

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arXiv2019

Making $K_{r+1}$-Free Graphs $r$-partite

The Erdős-Simonovits stability theorem states that for all ε>0 there exists α>0 such that if G is a K_{r+1}-free graph on n vertices with e(G) > ex(n,K_{r+1}) - αn^2, then one can remove εn^2 edges from G to obtain an r-partite graph. Füredi gave a short proof that one can choose α=ε. We give a bound for the relationship of αand \varepsilon which is asymptotically sharp as ε\to 0.

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arXiv2019

The typical structure of Gallai colorings and their extremal graphs

An edge coloring of a graph $G$ is a Gallai coloring if it contains no rainbow triangle. We show that the number of Gallai $r$-colorings of $K_n$ is $\left(\binom{r}{2}+o(1)\right)2^{\binom{n}{2}}$. This result indicates that almost all Gallai $r$-colorings of $K_n$ use only 2 colors. We also study the extremal behavior of Gallai $r$-colorings among all $n$-vertex graphs. We prove that the complete graph $K_n$ admits the largest number of Gallai $3$-colorings among all $n$-vertex graphs when $n$ is sufficiently large, while for $r\geq 4$, it is the complete bipartite graph $K_{\lfloor n/2 \rfloor, \lceil n/2 \rceil}$. Our main approach is based on the hypergraph container method, developed independently by Balogh, Morris, and Samotij as well as by Saxton and Thomason, together with some stability results for containers.

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arXiv2019

The maximum length of $K_r$-Bootstrap Percolation

Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollobás in 1968. In this process, we start with initial "infected" set of edges $E_0$, and we infect new edges according to a predetermined rule. Given a graph $H$ and a set of previously infected edges $E_t\subseteq E(K_n)$, we infect a non-infected edge $e$ if it completes a new copy of $H$ in $G=([n],E_t\cup e)$. A question raised by Bollobás asks for the maximum time the process can run before it stabilizes. Bollobás, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where $H=K_r$. They answered the question for $r\leq 4$ and gave a non-trivial lower bound for every $r\geq 5$. They also conjectured that the maximal running time is $o(n^2)$ for every integer $r$. In this paper we disprove their conjecture for every $r\geq 6$ and we give a better lower bound for the case $r=5$; in the proof we use the Behrend construction.

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arXiv2019

Structure and Supersaturation for Intersecting Families

The extremal problems regarding the maximum possible size of intersecting families of various combinatorial objects have been extensively studied. In this paper, we investigate supersaturation extensions, which in this context ask for the minimum number of disjoint pairs that must appear in families larger than the extremal threshold. We study the minimum number of disjoint pairs in families of permutations and in $k$-uniform set families, and determine the structure of the optimal families. Our main tool is a removal lemma for disjoint pairs. We also determine the typical structure of $k$-uniform set families without matchings of size $s$ when $n \ge 2sk + 38s^4$, and show that almost all $k$-uniform intersecting families on vertex set $[n]$ are trivial when $n\ge (2+o(1))k$.

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arXiv2019

Ordered Size Ramsey Number of Paths

An ordered graph is a simple graph with an ordering on its vertices. Define the ordered path $P_n$ to be the monotone increasing path with $n$ edges. The ordered size Ramsey number $\tilde{r}(P_r,P_s)$ is the minimum number $m$ for which there exists an ordered graph $H$ with $m$ edges such that every two-coloring of the edges of $H$ contains a red copy of $P_r$ or a blue copy of $P_s$. For $2\leq r\leq s$, we show $\frac{1}{8}r^2s\leq \tilde{r}(P_r,P_s)\leq Cr^2s(\log s)^3$, where $C>0$ is an absolute constant. This problem is motivated by the recent results of Bucić-Letzter-Sudakov and Letzter-Sudakov for oriented graphs.

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arXiv2019

Long monochromatic paths and cycles in 2-edge-colored multipartite graphs

We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i) cycle $C_{2n}$ with $2n$ vertices, (ii) cycle $C_{\geq 2n}$ with at least $2n$ vertices, (iii) path $P_{2n}$ with $2n$ vertices, and (iv) path $P_{2n+1}$ with $2n+1$ vertices. This implies a generalization for large $n$ of the conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi that for every $2$-edge-coloring of the complete $3$-partite graph $K_{n,n,n}$ there is a monochromatic path $P_{2n+1}$. An important tool is our recent stability theorem on monochromatic connected matchings.

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arXiv2019

Cubic graphs with small independence ratio

Let $i(r,g)$ denote the infimum of the ratio $\frac{α(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $α(G)$ is the independence number of $G$, and let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \ge 0.4375$, and Csóka improved it to $i(3,\infty) \ge 0.44533$ in 2016. Bollobás proved the upper bound $i(3,\infty) < \frac{6}{13}$ in 1981, and McKay improved it to $i(3,\infty) < 0.45537$ in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \le 0.454.$

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arXiv2019

The Ramsey Number of Fano Plane Versus Tight Path

The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer $N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. The Fano plane $\mathbb{F}$ is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that $R(H,\mathbb{F}) \ge 2(v(H)-1) + 1.$ Hypergraphs $H$ for which the equality holds are called $\mathbb{F}$-good. Conlon asked to determine all $H$ that are $\mathbb{F}$-good. In this short paper we make progress on this problem and prove that the tight path of length $n$ is $\mathbb{F}$-good.

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arXiv2018

On the number of linear hypergraphs of large girth

An $r$-uniform \textit{linear cycle} of length $\ell$, denoted by $C_{\ell}^r$, is an $r$-graph with edges $e_1, \ldots, e_{\ell}$ such that for every $i\in [\ell-1]$, $|e_i\cap e_{i+1}|=1$, $|e_{\ell}\cap e_1|=1$ and $e_i\cap e_j=\emptyset$ for all other pairs $\{i, j\},\ i\neq j$. For every $r\geq 3$ and $\ell\geq 4$, we show that there exists a constant $C$ depending on $r$ and $\ell$ such that the number of linear $r$-graphs of girth $\ell$ is at most $2^{Cn^{1+1/\lfloor \ell/2\rfloor}}$. Furthermore, we extend the result for $\ell=4$, proving that there exists a constant $C$ depending on $r$ such that the number of linear $r$-graphs without $C_{4}^r$ is at most $2^{Cn^{3/2}}$. The idea of the proof is to reduce the hypergraph enumeration problems to some graph enumeration problems, and then apply a variant of the graph container method, which may be of independent interest. We extend a breakthrough result of Kleitman and Winston on the number of $C_4$-free graphs, proving that the number of graphs containing at most $n^2/32\log^6 n$ $C_4$'s is at most $2^{11n^{3/2}}$, for sufficiently large $n$. We further show that for every $r\geq 3$ and $\ell\geq 2$, the number of graphs such that each of its edges is contained in only $O(1)$ cycles of length at most $2\ell$, is bounded by $2^{3(\ell+1)n^{1+1/\ell}}$ asymptotically.

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arXiv2018

Closing in on Hill's conjecture

Borrowing László Székely's lively expression, we show that Hill's conjecture is "asymptotically at least 98.5% true". This long-standing conjecture states that the crossing number cr($K_n$) of the complete graph $K_n$ is $H(n) := \frac{1}{4}\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor \lfloor \frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$, for all $n\ge 3$. This has been verified only for $n\le 12$. Using flag algebras, Norin and Zwols obtained the best known asymptotic lower bound for the crossing number of complete bipartite graphs, from which it follows that for every sufficiently large $n$, cr$(K_n) > 0.905\, H(n)$. Also using flag algebras, we prove that asymptotically cr$(K_n)$ is at least $0.985\, H(n)$. We also show that the spherical geodesic crossing number of $K_n$ is asymptotically at least $0.996\, H(n)$.

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arXiv2018

Packing chromatic number of subdivisions of cubic graphs

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $χ_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge. The questions on the value of the maximum of $χ_p(G)$ and of $χ_p(D(G))$ over the class of subcubic graphs $G$ appear in several papers. Gastineau and Togni asked whether $χ_p(D(G))\leq 5$ for any subcubic $G$, and later Bresar, Klavzar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that $χ_p(G)$ is not bounded in the class of subcubic graphs $G$. In contrast, in this paper we show that $χ_p(D(G))$ is bounded in this class, and does not exceed $8$.

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arXiv2018

Monochromatic Hilbert cubes and arithmetic progressions

The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic arithmetic progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic affine $k$--cube, that is, a set of the form$$\left\{x_0 + \sum_{b \in B} b : B \subseteq A\right\}$$ for some $|A|=k$ and $x_0 \in \mathbb{Z}$. We show the following relation between the Hilbert cube number and the Van der Waerden number. Let $k \geq 3$ be an integer. Then for every $ε>0$, there is a $c > 0$ such that $$h(k,4) \ge \min\{W(\lfloor c k^2\rfloor, 2), 2^{k^{2.5-ε}}\}.$$ Thus we improve upon state of the art lower bounds for $h(k,4)$ conditional on $W(k,2)$ being significantly larger than $2^k$. In the other direction, this shows that the if the Hilbert cube number is close its state of the art lower bounds, then $W(k,2)$ is at most doubly exponential in $k$. We also show the optimal result that for any Sidon set $A \subset \mathbb{Z}$, one has $$\left|\left\{\sum_{b \in B} b : B \subseteq A\right\}\right| = Ω( |A|^3) .$$

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arXiv2018

Sharp bound on the number of maximal sum-free subsets of integers

Cameron and Erdős asked whether the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of maximal sum-free sets. Here, we prove the following: For each $1\leq i \leq 4$, there is a constant $C_i$ such that, given any $n\equiv i \mod 4$, $\{1, \dots , n\}$ contains $(C_i+o(1)) 2^{n/4}$ maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, a structural result of Deshouillers, Freiman, Sós and Temkin and a recent bound on the number of subsets of integers with small sumset by Green and Morris. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.

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arXiv2018

Tilings in randomly perturbed dense graphs

A perfect $H$-tiling in a graph $G$ is a collection of vertex-disjoint copies of a graph $H$ in $G$ that together cover all the vertices in $G$. In this paper we investigate perfect $H$-tilings in a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds $m$ random edges to it. Specifically, for any fixed graph $H$, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect $H$-tiling with high probability. Our proof utilises Szemerédi's Regularity lemma as well as a special case of a result of Komlós concerning almost perfect $H$-tilings in dense graphs.

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