arXiv subjects

János Barát

Publications and source records attributed to János Barát.

36 records · Page 2Linked to original sources

arXiv2020

Intersecting and $2$-intersecting hypergraphs with maximal covering number: the Erdős-Lovász theme revisited

Erdős and Lovász noticed that an $r$-uniform intersecting hypergraph $H$ with maximal covering number, that is $τ(H)=r$, must have at least $\frac{8}{3}r-3$ edges. There has been no improvement on this lower bound for 45 years. We try to understand the reason by studying some small cases to see whether the truth lies very close to this simple bound. Let $q(r)$ denote the minimum number of edges in an intersecting $r$-uniform hypergraph. It was known that $q(3)=6$ and $q(4)=9$. We obtain the following new results: The extremal example for uniformity 4 is unique. Somewhat surprisingly it is not symmetric by any means. For uniformity 5, $q(5)=13$, and we found 3 examples, none of them being some known graph. We use both theoretical arguments and computer searches. In the footsteps of Erdős and Lovász, we also consider the special case, when the hypergraph is part of a finite projective plane. We determine the exact answer for $r\in \{3,4,5,6\}$. For uniformity 6, there is a unique extremal example. In a related question, we try to find $2$-intersecting $r$-uniform hypergraphs with maximal covering number, that is $τ(H)=r-1$. An infinite family of examples is to take all possible $r$-sets of a $(2r-2)$-vertex set. There is also a geometric candidate: biplanes. These are symmetric 2-designs with $λ=2$. We determined that only 3 biplanes of the 18 known examples are extremal.

math.CO↗

arXiv2020

Improvement on the crossing number of crossing-critical graphs

The crossing number of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. A graph $G$ is $k$-crossing-critical if its crossing number is at least $k$, but if we remove any edge of $G$, its crossing number drops below $k$. There are examples of $k$-crossing-critical graphs that do not have drawings with exactly $k$ crossings. Richter and Thomassen proved in 1993 that if $G$ is $k$-crossing-critical, then its crossing number is at most $2.5k+16$. We improve this bound to $2k+6\sqrt{k}+44$.

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arXiv2019

Decomposition of cubic graphs related to Wegner's conjecture

Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree $1$ (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least $1$ and contains no $3$-edge path. We prove the conjecture for Generalized Petersen graphs. We indicate that a coloring with the same properties might exist for any subcubic graph. We confirm this statement for all subcubic trees.

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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.

math.CO↗

arXiv2017

Transversals in generalized Latin squares

We are seeking a sufficient condition that forces a transversal in a generalized Latin square. A generalized Latin square of order $n$ is equivalent to a proper edge-coloring of $K_{n,n}$. A transversal corresponds to a multicolored perfect matching. Akbari and Alipour defined $l(n)$ as the least integer such that every properly edge-colored $K_{n,n}$, which contains at least $l(n)$ different colors, admits a multicolored perfect matching. They conjectured that $l(n)\leq n^2/2$ if $n$ is large enough. In this note we prove that $l(n)$ is bounded from above by $0.75n^2$ if $n>1$. We point out a connection to anti-Ramsey problems. We propose a conjecture related to a well-known result by Woolbright and Fu, that every proper edge-coloring of $K_{2n}$ admits a multicolored $1$-factor.

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arXiv2016

Rainbow matchings in bipartite multigraphs

Suppose that $k$ is a non-negative integer and a bipartite multigraph $G$ is the union of $$N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1)$$ matchings $M_1,\dots,M_N$, each of size $n$. We show that $G$ has a rainbow matching of size $n-k$, i.e. a matching of size $n-k$ with all edges coming from different $M_i$'s. Several choices of parameters relate to known results and conjectures.

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arXiv2015

Improvements on the density of maximal 1-planar graphs

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. Brandenburg et al. showed that there are maximal 1-planar graphs with only $\frac{45}{17}n + O(1)\approx 2.647n$ edges and maximal 1-plane graphs with only $\frac{7}{3}n+O(1)\approx 2.33n$ edges. On the other hand, they showed that a maximal 1-planar graph has at least $\frac{28}{13}n-O(1)\approx 2.15n-O(1)$ edges, and a maximal 1-plane graph has at least $2.1n-O(1)$ edges. We improve both lower bounds to $\frac{20n}{9}\approx 2.22n$.

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arXiv2015

Multipartite hypergraphs achieving equality in Ryser's conjecture

A famous conjecture of Ryser is that in an $r$-partite hypergraph the covering number is at most $r-1$ times the matching number. If true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has covering number at least $r-1$. We find that $f(r)$ is achieved only by linear hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}$. We also improve the general lower bound on $f(r)$, showing that $f(r)\ge 3.052r+O(1)$. We show that a stronger form of Ryser's conjecture that was used to prove the $r=3$ case fails for all $r>3$. We also prove a fractional version of the following stronger form of Ryser's conjecture: in an $r$-partite hypergraph there exists a set $S$ of size at most $r-1$, contained either in one side of the hypergraph or in an edge, whose removal reduces the matching number by 1.

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arXiv2012

Empty pentagons in point sets with collinearities

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential.

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arXiv2012

Edge-decomposition of graphs into copies of a tree with four edges

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Barát and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a 191-edge-connected graph of size divisible by 4, then $G$ has a $Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star.

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arXiv2011

Disproof of the List Hadwiger Conjecture

The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$.

math.CO↗

arXiv2011

Vertex coloring of plane graphs with nonrepetitive boundary paths

A sequence $s_1,s_2,...,s_k,s_1,s_2,...,s_k$ is a repetition. A sequence $S$ is nonrepetitive, if no subsequence of consecutive terms of $S$ form a repetition. Let $G$ be a vertex colored graph. A path of $G$ is nonrepetitive, if the sequence of colors on its vertices is nonrepetitive. If $G$ is a plane graph, then a facial nonrepetitive vertex coloring of $G$ is a vertex coloring such that any facial path is nonrepetitive. Let $π_f(G)$ denote the minimum number of colors of a facial nonrepetitive vertex coloring of $G$. Jendro\vl and Harant posed a conjecture that $π_f(G)$ can be bounded from above by a constant. We prove that $π_f(G)\le 24$ for any plane graph $G$.

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arXiv2010

Large B_d-free and union-free subfamilies

For a property $Γ$ and a family of sets $\cF$, let $f(\cF,Γ)$ be the size of the largest subfamily of $\cF$ having property $Γ$. For a positive integer $m$, let $f(m,Γ)$ be the minimum of $f(\cF,Γ)$ over all families of size $m$. A family $\cF$ is said to be $B_d$-free if it has no subfamily $\cF'=\{F_I: I \subseteq [d]\}$ of $2^d$ distinct sets such that for every $I,J \subseteq [d]$, both $F_I \cup F_J=F_{I \cup J}$ and $F_I \cap F_J = F_{I \cap J}$ hold. A family $\cF$ is $a$-union free if $F_1\cup ... F_a \neq F_{a+1}$ whenever $F_1,..,F_{a+1}$ are distinct sets in $\FF$. We verify a conjecture of Erd\H os and Shelah that $f(m, B_2\text{\rm -free})=Θ(m^{2/3})$. We also obtain lower and upper bounds for $f(m, B_d\text{\rm -free})$ and $f(m,a\text{\rm -union free})$.

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arXiv2009

On the structure of graphs with path-width at most two

Nancy G. Kinnersley and Michael A. Langston has determined the excluded minors for the class of graphs with path-width at most two by computer. Their list consisted of 110 graphs. Such a long list is difficult to handle and gives no insight to structural properties. We take a different route, and concentrate on the building blocks and how they are glued together. In this way, we get a characterization of 2-connected and 2-edge-connected graphs with path-width at most two. Along similar lines, we sketch the complete characterization of graphs with path-width at most two.

math.CO↗

arXiv2009

Elementary proof techniques for the maximum number of islands

Islands are combinatorial objects that can be intuitively defined on a board consisting of a finite number of cells. Based on the neighbor relation of the cells, it is a fundamental property that two islands are either containing or disjoint. Recently, numerous extremal questions have been answered using different methods. We show elementary techniques unifying these approaches. Our building parts are based on rooted binary trees and discrete geometry. Among other things, we show the maximum cardinality of islands on a toroidal board and in a hypercube. We also strengthen a previous result by rarefying the neighborhood relation.

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arXiv2009

On winning fast in Avoider-Enforcer games

We analyze the duration of the unbiased Avoider-Enforcer game for three basic positional games. All the games are played on the edges of the complete graph on $n$ vertices, and Avoider's goal is to keep his graph outerplanar, diamond-free and $k$-degenerate, respectively. It is clear that all three games are Enforcer's wins, and our main interest lies in determining the largest number of moves Avoider can play before losing. Extremal graph theory offers a general upper bound for the number of Avoider's moves. As it turns out, for all three games we manage to obtain a lower bound that is just an additive constant away from that upper bound. In particular, we exhibit a strategy for Avoider to keep his graph outerplanar for at least $2n-8$ moves, being just 6 short of the maximum possible. A diamond-free graph can have at most $d(n)=\lceil\frac{3n-5}{2}\rceil$ edges, and we prove that Avoider can play for at least $d(n)-3$ moves. Finally, if $k$ is small compared to $n$, we show that Avoider can keep his graph $k$-degenerate for as many as $e(n)$ moves, where $e(n)$ is the maximum number of edges a $k$-degenerate graph can have.

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arXiv2009

Towards The Albertson Conjecture

Albertson conjectured that if a graph $G$ has chromatic number $r$ then its crossing number is at least as much as the crossing number of $K_r$. Albertson, Cranston, and Fox verified the conjecture for $r\le 12$. We prove the statement for $r\le 16$.

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arXiv2007

Notes on Nonrepetitive Graph Colouring

A vertex colouring of a graph is \emph{nonrepetitive on paths} if there is no path $v_1,v_2,...,v_{2t}$ such that v_i and v_{t+i} receive the same colour for all i=1,2,...,t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree $Δ$ has a $f(Δ)$-colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree $Δ$ has a $O(kΔ)$-colouring that is nonrepetitive on paths, and a $O(kΔ^3)$-colouring that is nonrepetitive on walks.

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