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Jan Hladký

Publications and source records attributed to Jan Hladký.

At least 37 records · Page 2Linked to original sources
arXiv2020

A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems

Given a measurable set $A\subset \mathbb R^d$ we consider the "large-distance graph" $\mathcal{G}_A$, on the ground set $A$, in which each pair of points from $A$ whose distance is bigger than 2 forms an edge. We consider the problems of maximizing the $2d$-dimensional Lebesgue measure of the edge set as well as the $d$-dimensional Lebesgue measure of the vertex set of a large-distance graph in the $d$-dimensional Euclidean space that contains no copies of a complete graph on $k$ vertices. The former problem may be seen as a continuous analogue of Turán's classical graph theorem, and the latter as a graph-theoretic analogue of the classical isodiametric problem. Our main result yields an analogue of Mantel's theorem for large-distance graphs. Our approach employs an isodiametric inequality in an annulus, which might be of independent interest.

math.CO↗
arXiv2019

Two remarks on graph norms

For a graph $H$, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions $W$ in $L^p$, $p\geq e(H)$, denoted by $t(H,W)$. One may then define corresponding functionals $\|W\|_{H}:=|t(H,W)|^{1/e(H)}$ and $\|W\|_{r(H)}:=t(H,|W|)^{1/e(H)}$ and say that $H$ is (semi-)norming if $\|.\|_{H}$ is a (semi-)norm and that $H$ is weakly norming if $\|.\|_{r(H)}$ is a norm. We obtain two results that contribute to the theory of (weakly) norming graphs. Firstly, answering a question of Hatami, who estimated the modulus of convexity and smoothness of $\|.\|_{H}$, we prove that $\|.\|_{r(H)}$ is not uniformly convex nor uniformly smooth, provided that $H$ is weakly norming. Secondly, we prove that every graph $H$ without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of $H$ when studying graph norms. In particular, we correct an error in the original statement of the aforementioned theorem by Hatami.

math.CO↗
arXiv2019

Independent sets, cliques, and colorings in graphons

We study graphon counterparts of the chromatic and the clique number, the fractional chromatic number, the b-chromatic number, and the fractional clique number. We establish some basic properties of the independence set polytope in the graphon setting, and duality properties between the fractional chromatic number and the fractional clique number. We present a notion of perfect graphons and characterize them in terms of induced densities of odd cycles and its complements.

math.CO↗
arXiv2018

Komlós's tiling theorem via graphon covers

Komlos [Komlos: Tiling Turan Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum-degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H, thus essentially extending the Hajnal-Szemeredi theorem which deals with the case when H is a clique. We give a proof of a graphon version of Komlos's theorem. To prove this graphon version, and also to deduce from it the original statement about finite graphs, we use the machinery introduced in [Hladky, Hu, Piguet: Tilings in graphons, arXiv:1606.03113]. We further prove a stability version of Komlos's theorem.

math.CO↗
arXiv2018

The local limit of the uniform spanning tree on dense graphs

Let $G$ be a connected graph in which almost all vertices have linear degrees and let $T$ be a uniform spanning tree of $G$. For any fixed rooted tree $F$ of height $r$ we compute the asymptotic density of vertices $v$ for which the $r$-ball around $v$ in $T$ is isomorphic to $F$. We deduce from this that if $\{G_n\}$ is a sequence of such graphs converging to a graphon $W$, then the uniform spanning tree of $G_n$ locally converges to a multi-type branching process defined in terms of $W$. As an application, we prove that in a graph with linear minimum degree, with high probability, the density of leaves in a uniform spanning tree is at least $1/e-o(1)$, the density of vertices of degree $2$ is at most $1/e+o(1)$ and the density of vertices of degree $k\geq 3$ is at most ${(k-2)^{k-2} \over (k-1)! e^{k-2}} + o(1)$. These bounds are sharp.

math.PR↗
arXiv2016

The Approximate Loebl-Komlós-Sós Conjecture III: The finer structure of LKS graphs

This is the third of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $α>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+α)n$ vertices of degree at least $(1+α)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics. In the second paper, we found a combinatorial structure inside the decomposition. In this paper, we will give a refinement of this structure. In the forthcoming fourth paper, the refined structure will be used for embedding the tree $T$.

math.CO↗
arXiv2016

The approximate Loebl-Komlós-Sós Conjecture IV: Embedding techniques and the proof of the main result

This is the last paper of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $α>0$ there exists a number~$k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+α)n$ vertices of degree at least $(1+α)k$ contains each tree $T$ of order $k$ as a subgraph. In the first two papers of this series, we decomposed the host graph $G$, and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure, and proved that any graph satisfying the conditions of the above approximate version of the Loebl-Komlos-Sos Conjecture contains one of ten specific configurations. In this paper we embed the tree $T$ in each of the ten configurations.

math.CO↗
arXiv2016

Cliques in dense inhomogeneous random graphs

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of G(n,W) for each fixed n, and give examples of graphons for which G(n,W) exhibits wild long-term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent interest: Every graphon avoiding a fixed graph is countably-partite.

math.CO↗
arXiv2016

Rank of divisors on tropical curves

We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, an elementary proof of the Riemann-Roch theorem for tropical curves, similar to the recent proof of the Riemann-Roch theorem for graphs by Baker and Norine, is presented. In addition, a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to the rank of D on the corresponding metric graph is confirmed, and an algorithm for computing the rank of a divisor on a tropical curve is constructed.

math.CO↗
arXiv2016

The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs

This is the second of a series of four papers in which we prove the following relaxation of the Loebl-Komlos--Sos Conjecture: For every $α>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+α)n$ vertices of degree at least $(1+α)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics; this decomposition might be viewed as an analogue of a regular partition for sparse graphs. In the present paper, we find a combinatorial structure inside this decomposition. In the last two papers, we refine the structure and use it for embedding the tree $T$.

math.CO↗
arXiv2015

The approximate Loebl-Komlós-Sós Conjecture I: The sparse decomposition

In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $α>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+α)n$ vertices of degree at least $(1+α)k$ contains each tree $T$ of order $k$ as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemerédi regularity lemma: we decompose the graph $G$, find a suitable combinatorial structure inside the decomposition, and then embed the tree $T$ into $G$ using this structure. Since for sparse graphs $G$, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a combinatorial structure suitable inside the decomposition, which we then use for embedding the tree.

math.CO↗
arXiv2015

The approximate Loebl-Komlós-Sós Conjecture

We prove the following version of the Loebl-Komlos-Sos Conjecture: For every alpha>0 there exists a number M such that for every k>M every n-vertex graph G with at least (0.5+alpha)n vertices of degree at least (1+alpha)k contains each tree T of order k as a subgraph. The method to prove our result follows a strategy common to approaches which employ the Szemeredi Regularity Lemma: we decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. However, the decomposition given by the Regularity Lemma is not of help when G is sparse. To surmount this shortcoming we use a more general decomposition technique: each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the Regularity Lemma), and two other objects each exhibiting certain expansion properties.

math.CO↗
arXiv2014

An approximate version of the Tree Packing Conjecture

We prove that for any pair of constants $ε>0$ and $Δ$ and for $n$ sufficiently large, every family of trees of orders at most $n$, maximum degrees at most $Δ$, and with at most $\binom{n}{2}$ edges in total packs into $K_{(1+ε)n}$. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

math.CO↗
arXiv2014

An extension of Turán's Theorem, uniqueness and stability

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.

math.CO↗
arXiv2014

A density Corrádi-Hajnal Theorem

We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph which does not contain $k+1$ vertex-disjoint triangles. This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corradi-Hajnal Theorem.

math.CO↗
arXiv2014

Hamilton cycles in dense vertex-transitive graphs

A famous conjecture of Lovász states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.

math.CO↗
arXiv2012

On the Number of Pentagons in Triangle-Free Graphs

Using the formalism of flag algebras, we prove that every triangle-free graph $G$ with $n$ vertices contains at most $(n/5)^5$ cycles of length five. Moreover, the equality is attained only when $n$ is divisible by five and $G$ is the balanced blow-up of the pentagon. We also compute the maximal number of pentagons and characterize extremal graphs in the non-divisible case provided $n$ is sufficiently large. This settles a conjecture made by Erdős in 1984.

math.CO↗