arXiv subjects

Tibor Szabó

Publications and source records attributed to Tibor Szabó.

35 records · Page 2Linked to original sources

arXiv2019

Enumerating extensions of mutually orthogonal Latin squares

Two $n \times n$ Latin squares $L_1, L_2$ are said to be orthogonal if, for every ordered pair $(x,y)$ of symbols, there are coordinates $(i,j)$ such that $L_1(i,j) = x$ and $L_2(i,j) = y$. A $k$-MOLS is a sequence of $k$ pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed $k$, log-asymptotically tight bounds on the number of $k$-MOLS. To study the situation when $k$ grows with $n$, we bound the number of ways a $k$-MOLS can be extended to a $(k+1)$-MOLS. These bounds are again tight for constant $k$, and allow us to deduce upper bounds on the total number of $k$-MOLS for all $k$. These bounds are close to tight even for $k$ linear in $n$, and readily generalize to the broader class of gerechte designs, which include Sudoku squares.

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arXiv2019

Singer difference sets and the projective norm graph

We demonstrate a close connection between the classic planar Singer difference sets and certain norm equation systems arising from projective norm graphs. This, on the one hand leads to a novel description of planar Singer difference sets as a subset $\mathcal{H}$ of $\mathcal{N}$, the group of elements of norm 1 in the field extension $\mathbb{F}_{q^3}/\mathbb{F}_q$. $\mathcal{H}$ is given as the solution set of a simple polynomial equation, and we obtain an explicit formula expressing each non-identity element of $\mathcal{N}$ as a product $B\cdot C^{-1}$ with $B, C\in \mathcal{H}$. The description and the definitions naturally carry over to the nonplanar and the infinite setting. On the other hand, relying heavily on the difference set properties, we also complete the proof that the projective norm graph $\text{NG}(q,4)$ does contain the complete bipartite graph $K_{4,6}$ for every prime power $q \geq 5$. This complements the property, known for more than two decades, that projective norm graphs do not contain $K_{4,7}$ (and hence provide tight lower bounds for the Turán number $ex(n,K_{4,7})$).

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arXiv2019

Exploring Projective Norm Graphs

The projective norm graphs $\text{NG}(q,t)$ provide tight constructions for the Turán number of complete bipartite graphs $K_{t,s}$ with $s>(t-1)!$. In this paper we determine their automorphism group and explore their small subgraphs. To this end we give quite precise estimates on the number of solutions of certain equation systems involving norms over finite fields. The determination of the largest integer $s_t$, such that the projective norm graph $\text{NG}(q,t)$ contains $K_{t,s_t}$ for all large enough prime powers $q$ is an important open question with far-reaching general consequences. The best known bounds, $t-1\leq s_t \leq (t-1)!$, are far apart for $t\geq 4$. Here we prove that $\text{NG}(q,4)$ does contain (many) $K_{4,6}$ for any prime power $q$ not divisble by $2$ or $3$. This greatly extends recent work of Grosu, using a completely different approach. Along the way we also count the copies of any fixed $3$-degenerate subgraph, and find that projective norm graphs are quasirandom with respect to this parameter. Some of these results also extend the work of Alon and Shikhelman on generalized Turán numbers. Finally we also give a new, more elementary proof for the $K_{4,7}$-freeness of $\text{NG}(q,4)$.

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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

On the Odd Cycle Game and Connected Rules

We study the positional game where two players, Maker and Breaker, alternately select respectively $1$ and $b$ previously unclaimed edges of $K_n$. Maker wins if she succeeds in claiming all edges of some odd cycle in $K_n$ and Breaker wins otherwise. Improving on a result of Bednarska and Pikhurko, we show that Maker wins the odd cycle game if $b \leq ((4 - \sqrt{6})/5 + o(1)) n$. We furthermore introduce "connected rules" and study the odd cycle game under them, both in the Maker-Breaker as well as in the Client-Waiter variant.

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arXiv2018

On the optimality of the uniform random strategy

The concept of biased Maker-Breaker games, introduced by Chvátal and Erd{\H o}s, is a central topic in the field of positional games, with deep connections to the theory of random structures. For any given hypergraph ${\cal H}$ the main questions is to determine the smallest bias $q({\cal H})$ that allows Breaker to force that Maker ends up with an independent set of ${\cal H}$. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies a couple of natural `container-type' regularity conditions about the degree of subsets of its vertices. This will enable us to derive a hypergraph generalization of the $H$-building games, studied for graphs by Bednarska and Łuczak. Furthermore, we investigate the biased version of generalizations of the van der Waerden games introduced by Beck. We refer to these generalizations as Rado games and determine their threshold bias up to constant factors by applying our general criteria. We find it quite remarkable that a purely game theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, when the generalizations to hypergraphs in the analogous setup of sparse random discrete structures are usually quite challenging.

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arXiv2018

A family of extremal hypergraphs for Ryser's conjecture

Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph, the vertex cover number is at most $r{-}1$ times the matching number. This conjecture is only known to be true for $r\leq 3$ in general and for $r\leq 5$ if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. $r$-partite hypergraphs whose cover number is $r-1$ times its matching number. Aside from a few sporadic examples, the set of uniformities $r$ for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order $r-1$ exists. We produce a new infinite family of $r$-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order $r-2$ exists. Our construction is flexible enough to produce a large number of non-isomorphic extremal hypergraphs. In particular, we define what we call the {\em Ryser poset} of extremal intersecting $r$-partite $r$-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in $\sqrt{r}$. This provides further evidence for the difficulty of Ryser's Conjecture.

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arXiv2016

Extremal Hypergraphs for Ryser's Conjecture: Connectedness of Line Graphs of Bipartite Graphs

In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the lower bound $\frac{\dim(\mathcal{I}(G))}{2} - 2$ on the connectedness of the independence complex $\mathcal{I}(G)$ of line graphs of bipartite graphs $G$ is tight. In our main theorem we characterize the extremal examples. Our proof of this characterization is based on topological machinery. Our motivation for studying this problem comes from a classical conjecture of Ryser. Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $(r - 1)$-times the size of the largest matching. For $r = 2$, the conjecture is simply König's Theorem. It has also been proven for $r = 3$ by Aharoni using a beautiful topological argument. In a separate paper we characterize the extremal examples for the $3$-uniform case of Ryser's Conjecture (i.e., Aharoni's Theorem), and in particular resolve an old conjecture of Lovász for the case of Ryser-extremal $3$-graphs. Our main result in this paper will provide us with valuable structural information for that characterization. Its proof is based on the observation that link graphs of Ryser-extremal $3$-uniform hypergraphs are exactly the bipartite graphs we study here.

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arXiv2016

Extremal Hypergraphs for Ryser's Conjecture: Home-Base Hypergraphs

Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $r - 1$ times the size of the largest matching. For $r = 2$, the conjecture is simply König's Theorem and every bipartite graph is a witness for its tightness. The conjecture has also been proven for $r = 3$ by Aharoni using topological methods, but the proof does not give information on the extremal $3$-uniform hypergraphs. Our goal in this paper is to characterize those hypergraphs which are tight for Aharoni's Theorem. Our proof of this characterization is also based on topological machinery, particularly utilizing results on the (topological) connectedness of the independence complex of the line graph of the link graphs of $3$-uniform Ryser-extremal hypergraphs, developed in a separate paper. The current paper contains the second, structural hypergraph-theoretic part of the argument, where we use the information on the line graph of the link graphs to nail down the elements of a structure we call \emph{home-base hypergraph}. While there is a single minimal home-base hypergraph with matching number $k$ for every positive integer $k \in \mathbb{N}$, home-base hypergraphs with matching number $k$ are far from being unique. There are infinitely many of them and each of them is composed of $k$ copies of two different kinds of basic structures, whose hyperedges can intersect in various restricted, but intricate ways. Our characterization also proves an old and wide open strengthening of Ryser's Conjecture, due to Lovász, for the $3$-uniform extremal case, that is, for hypergraphs with $τ= 2 ν$.

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arXiv2016

Sharp thresholds for half-random games I

We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this paper we consider the scenario when Maker plays randomly and Breaker is "clever", and determine the sharp threshold bias of classical graph games, such as connectivity, Hamiltonicity, and minimum degree-$k$. We treat the other case, that is when Breaker plays randomly, in a separate paper. The traditional, deterministic version of these games, with two optimal players playing, are known to obey the so-called probabilistic intuition. That is, the threshold bias of these games is asymptotically equal to the threshold bias of their random counterpart, where players just take edges uniformly at random. We find, that despite this remarkably precise agreement of the results of the deterministic and the random games, playing randomly against an optimal opponent is not a good idea: the threshold bias becomes significantly more tilted towards the random player. An important qualitative aspect of the probabilistic intuition carries through nevertheless: the bottleneck for Maker to occupy a connected graph is still the ability to avoid isolated vertices in her graph.

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arXiv2016

Sharp thresholds for half-random games II

We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this work we focus on the case of Breaker playing randomly and Maker being "clever". The reverse scenario is treated in a separate paper. We determine the sharp threshold bias of classical games played on the edge set of the complete graph $K_n$, such as connectivity, perfect matching, Hamiltonicity, and minimum degree-$1$. In all of these games, the threshold is equal to the trivial upper bound implied by the number of edges needed for Maker to occupy a winning set. Moreover, we show that the clever Maker can not only win against an asymptotically optimal bias, but can do so very fast, wasting only logarithmically many moves (while the winning set sizes are linear in $n$).

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arXiv2015

On the Concentration of the Domination Number of the Random Graph

In this paper we study the behaviour of the domination number of the Erdős-Rényi random graph $\mathcal{G}(n,p)$. Extending a result of Wieland and Godbole we show that the domination number of $\mathcal{G}(n,p)$ is equal to one of two values asymptotically almost surely whenever $p \gg \frac{\ln^2n}{\sqrt{n}}$. The explicit values are exactly at the first moment threshold, that is where the expected number of dominating sets starts to tend to infinity. For small $p$ we also provide various non-concentration results which indicate why some sort of lower bound on the probability $p$ is necessary in our first theorem. Concentration, though not on a constant length interval, is proven for every $p\gg 1/n$. These results show that unlike in the case of $p \gg \frac{\ln^2n}{\sqrt{n}}$ where concentration of the domination number happens around the first moment threshold, for $p = O(\ln n/n)$ it does so around the median. In particular, in this range the two are far apart from each other.

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arXiv2014

Graphs without proper subgraphs of minimum degree 3 and short cycles

We study graphs on $n$ vertices which have $2n-2$ edges and no proper induced subgraphs of minimum degree $3$. Erdős, Faudree, Gyárfás, and Schelp conjectured that such graphs always have cycles of lengths $3,4,5,\dots, C(n)$ for some function $C(n)$ tending to infinity. We disprove this conjecture, resolve a related problem about leaf-to-leaf path lengths in trees, and characterize graphs with $n$ vertices and $2n-2$ edges, containing no proper subgraph of minimum degree $3$.

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arXiv2013

Conflict-free coloring of graphs

We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős-Rényi random graph G(n,p) and give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the conflict-free chromatic number differs from the domination number by at most 3.

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arXiv2012

How many colors guarantee a rainbow matching?

Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow $t$-matching in an r-partite r-uniform multi-hypergraph whose edges are colored with f colors such that every color class is a matching of size t. This problem was posed by Aharoni and Berger, who asked to determine the minimum number of colors which guarantees a rainbow matching. We improve on the known upper bounds for this problem for all values of the parameters. In particular for every fixed r, we give an upper bound which is polynomial in t, improving the superexponential estimate of Alon. Our proof also works in the setting not requiring the hypergraph to be r-partite.

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arXiv2011

On covering expander graphs by Hamilton cycles

The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree $Δ$ satisfies some basic expansion properties and contains a family of $(1-o(1))Δ/2$ edge disjoint Hamilton cycles, then there also exists a covering of its edges by $(1+o(1))Δ/2$ Hamilton cycles. This implies that for every $α>0$ and every $p \geq n^{α-1}$ there exists a covering of all edges of $G(n,p)$ by $(1+o(1))np/2$ Hamilton cycles asymptotically almost surely, which is nearly optimal.

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arXiv2008

Fast winning strategies in Avoider-Enforcer games

In numerous positional games the identity of the winner is easily determined. In this case one of the more interesting questions is not {\em who} wins but rather {\em how fast} can one win. These type of problems were studied earlier for Maker-Breaker games; here we initiate their study for unbiased Avoider-Enforcer games played on the edge set of the complete graph $K_n$ on $n$ vertices. For several games that are known to be an Enforcer's win, we estimate quite precisely the minimum number of moves Enforcer has to play in order to win. We consider the non-planarity game, the connectivity game and the non-bipartite game.

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