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Jan Volec

Publications and source records attributed to Jan Volec.

At least 37 records · Page 2Linked to original sources
arXiv2022

Towards characterizing locally common graphs

A graph H is common if the number of monochromatic copies of H in a 2-edge-coloring of the complete graph is asymptotically minimized by the random coloring. The classification of common graphs is one of the most intriguing problems in extremal graph theory. We study the notion of weakly locally common graphs considered by Csóka, Hubai and Lovász [arXiv:1912.02926], where the graph is required to be the minimizer with respect to perturbations of the random 2-edge-coloring. We give a complete analysis of the 12 initial terms in the Taylor series determining the number of monochromatic copies of H in such perturbations and classify graphs H based on this analysis into three categories: graphs of Class I are weakly locally common, graphs of Class II are not weakly locally common, and graphs of Class III cannot be determined to be weakly locally common or not based on the initial 12 terms. As a corollary, we obtain new necessary conditions on a graph to be common and new sufficient conditions on a graph to be not common.

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arXiv2020

Sharp bounds for decomposing graphs into edges and triangles

For a real constant $α$, let $π_3^α(G)$ be the minimum of twice the number of $K_2$'s plus $α$ times the number of $K_3$'s over all edge decompositions of $G$ into copies of $K_2$ and $K_3$, where $K_r$ denotes the complete graph on $r$ vertices. Let $π_3^α(n)$ be the maximum of $π_3^α(G)$ over all graphs $G$ with $n$ vertices. The extremal function $π_3^3(n)$ was first studied by Győri and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315--320]. In a recent progress on this problem, Král', Lidický, Martins and Pehova [Decomposing graphs into edges and triangles, Combin. Prob. Comput. 28 (2019) 465--472] proved via flag algebras that $π_3^3(n)\le (1/2+o(1))n^2$. We extend their result by determining the exact value of $π_3^α(n)$ and the set of extremal graphs for all $α$ and sufficiently large $n$. In particular, we show for $α=3$ that $K_n$ and the complete bipartite graph $K_{\lfloor n/2\rfloor,\lceil n/2\rceil}$ are the only possible extremal examples for large $n$.

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arXiv2020

Counterexamples to a conjecture of Harris on Hall ratio

The Hall ratio of a graph $G$ is the maximum value of $v(H) / α(H)$ taken over all non-null subgraphs $H$ of $G$. For any graph, the Hall ratio is a lower-bound on its fractional chromatic number. In this note, we present various constructions of graphs whose fractional chromatic number grows much faster than their Hall ratio. This refutes a conjecture of Harris.

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arXiv2019

Compactness and finite forcibility of graphons

Graphons are analytic objects associated with convergent sequences of graphs. Problems from extremal combinatorics and theoretical computer science led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible. Following the intuition that such graphons should have finitary structure, Lovasz and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space is not compact. The construction method gives a general framework for constructing finitely forcible graphons with non-trivial properties.

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arXiv2018

Limits of Order Types

We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erdős problem of finding the minimal density of 5-or 6-tuples in convex position in an arbitrary point set, and also an inequality expressing the difficulty of sampling order types uniformly. Next we establish results on the analytic representation of limits of order types by planar measures. Our main result is a rigidity theorem: we show that if sampling two measures induce the same probability distribution on order types, then these measures are projectively equivalent provided the support of at least one of them has non-empty interior. We also show that some condition on the Hausdorff dimension of the support is necessary to obtain projective rigidity and we construct limits of order types that cannot be represented by a planar measure. Returning to combinatorial geometry we relate the regularity of this analytic representation to the aforementioned problem of Erdős on the density of k-tuples in convex position, for large k.

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arXiv2018

Minimum number of edges that occur in odd cycles

If a graph has $n\ge4k$ vertices and more than $n^2/4$ edges, then it contains a copy of $C_{2k+1}$. In 1992, Erdős, Faudree and Rousseau showed even more, that the number of edges that occur in a triangle is at least $2\lfloor n/2\rfloor+1$, and this bound is tight. They also showed that the minimum number of edges that occur in a $C_{2k+1}$ for $k\ge2$ is at least $11n^2/144-O(n)$, and conjectured that for any $k\ge2$, the correct lower bound should be $2n^2/9-O(n)$. Very recently, Füredi and Maleki constructed a counterexample for $k=2$ and proved asymptotically matching lower bound, namely that for any $\varepsilon>0$ graphs with $(1+\varepsilon)n^2/4$ edges contain at least $(2+\sqrt{2})n^2/16 \approx 0.2134n^2$ edges that occur in $C_5$. In this paper, we use a different approach to tackle this problem and obtain the following stronger result: Any $n$-vertex graph with at least $\lfloor n^2/4\rfloor+1$ edges has at least $(2+\sqrt{2})n^2/16-O(n^{15/8})$ edges that occur in $C_5$. Next, for all $k\ge 3$ and $n$ sufficiently large, we determine the exact minimum number of edges that occur in $C_{2k+1}$ for $n$-vertex graphs with more than $n^2/4$ edges, and show it is indeed equal to $\lfloor\frac{n^2}4\rfloor+1-\lfloor\frac{n+4}6\rfloor\lfloor\frac{n+1}6\rfloor=2n^2/9-O(n)$. For both results, we give a structural description of the extremal configurations as well as obtain the corresponding stability results, which answer a conjecture of Füredi and Maleki. The main ingredient is a novel approach that combines the flag algebras together with ideas from finite forcibility of graph limits. This approach allowed us to keep track of the extra edge needed to guarantee an existence of a $C_{2k+1}$. Also, we establish the first application of semidefinite method in a setting, where the set of tight examples has exponential size, and arises from different constructions.

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arXiv2018

A bound on the inducibility of cycles

In 1975, Pippenger and Golumbic conjectured that every n-vertex graph has at most $n^k/(k^k - k)$ induced cycles of length k for k at least 5. We prove that every n-vertex graph has at most $2 n^k/k^k$ induced cycles of length k.

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arXiv2017

Densities of 3-vertex graphs

Let d_i(G) be the density of the 3-vertex i-edge graph in a graph G, i.e., the probability that three random vertices induce a subgraph with i edges. Let S be the set of all quadruples (d_0,d_1,d_2,d_3) that are arbitrary close to 3-vertex graph densities in arbitrary large graphs. Huang, Linial, Naves, Peled and Sudakov have recently determined the projection of the set S to the (d_0,d_3) plane. We determine the projection of the set S to all the remaining planes.

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arXiv2016

Bounded colorings of multipartite graphs and hypergraphs

Let $c$ be an edge-coloring of the complete $n$-vertex graph $K_n$. The problem of finding properly colored and rainbow Hamilton cycles in $c$ was initiated in 1976 by Bollobás and Erd\H os and has been extensively studied since then. Recently it was extended to the hypergraph setting by Dudek, Frieze and Ruciński. We generalize these results, giving sufficient local (resp. global) restrictions on the colorings which guarantee a properly colored (resp. rainbow) copy of a given hypergraph $G$. We also study multipartite analogues of these questions. We give (up to a constant factor) optimal sufficient conditions for a coloring $c$ of the complete balanced $m$-partite graph to contain a properly colored or rainbow copy of a given graph $G$ with maximum degree $Δ$. Our bounds exhibit a surprising transition in the rate of growth, showing that the problem is fundamentally different in the regimes $Δ\gg m$ and $Δ\ll m$ Our main tool is the framework of Lu and Székely for the space of random bijections, which we extend to product spaces.

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arXiv2016

Properly colored and rainbow copies of graphs with few cherries

Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph K_n such that at each vertex every color appears only constantly many times. In 1979, Shearer conjectured that such a coloring c must contain a properly colored copy of G. We establish this conjecture in a strong form, showing that it holds even for graphs G with O(n^(4/3)) cherries and moreover this bound on the number of cherries is best possible up to a constant factor. We also prove that one can find a rainbow copy of such G in every edge-coloring of K_n in which all colors appear bounded number of times. Our proofs combine a framework of Lu and Szekely for using the lopsided Lovasz local lemma in the space of random bijections together with some additional ideas.

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arXiv2014

Minimum number of monotone subsequences of length 4 in permutations

We show that for every sufficiently large $n$, the number of monotone subsequences of length four in a permutation on $n$ points is at least $\binom{\lfloor n/3 \rfloor}{4} + \binom{\lfloor(n+1)/3\rfloor}{4} + \binom{\lfloor (n+2)/3\rfloor}{4}$. Furthermore, we characterize all permutations on $[n]$ that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colorings of complete graphs with two colors, where the number of monochromatic $K_4$'s is minimized. We show that all the extremal colorings must contain monochromatic $K_4$'s only in one of the two colors. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

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arXiv2014

A problem of Erdos and Sos on 3-graphs

We show that for every positive epsilon there exist positive delta and n_0 such that every 3-uniform hypergraph on n>=n_0 vertices with the property that every k-vertex subset, where k>=delta*n, induces at least (1/4 + epsilon)*{k \choose 3} edges, contains K4- as a subgraph, where K4- is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdos and Sos. The constant 1/4 is the best possible.

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arXiv2014

Rainbow triangles in three-colored graphs

Erdos and Sos proposed a problem of determining the maximum number F(n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F(n) = F(a)+ F(b)+F(c)+F(d)+abc+abd+acd+bcd, where a+b+c+d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4k for all k. These results imply that lim F(n) n^3/6 = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments.

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arXiv2013

A note on acyclic vertex-colorings

We prove that the acyclic chromatic number of a graph with maximum degree $Δ$ is less than $2.835Δ^{4/3}+Δ$. This improves the previous upper bound, which was $50Δ^{4/3}$. To do so, we draw inspiration from works by Alon, McDiarmid and Reed and by Esperet and Parreau.

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arXiv2013

Extensions of Fractional Precolorings show Discontinuous Behavior

We study the following problem: given a real number k and integer d, what is the smallest epsilon such that any fractional (k+epsilon)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k+epsilon)-coloring of the whole graph? The exact values of epsilon were known for k=2 and k\ge3 and any d. We determine the exact values of epsilon for k \in (2,3) if d=4, and k \in [2.5,3) if d=6, and give upper bounds for k \in (2,3) if d=5,7, and k \in (2,2.5) if d=6. Surprisingly, epsilon viewed as a function of k is discontinuous for all those values of d.

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arXiv2012

Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs

We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an edge-cut of size at least 1.33008n, where n is the number of vertices of G, and has fractional cut covering number at most 1.127752. The lower bound on the size of maximum edge-cut also applies to random cubic graphs. Specifically, a random n-vertex cubic graph a.a.s. contains an edge cut of size 1.33008n.

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