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Fractionally Isomorphic Graphs and Graphons
Hladký, Jan ; Hng, Eng Keat
Fractional isomorphism is a well-studied relaxation of graph isomorphism with a very rich theory. Grebík and Rocha [Combinatorica 42, pp 365–404 (2022)] developed a concept of fractional isomorphism for graphons and proved that it enjoys an analogous theory. In particular, they proved that if two sequences of graphs that are fractionally isomorphic converge to two graphons, then these graphons are fractionally isomorphism. Answering the main question from ibid, we prove the converse of the statement above: If we have two fractionally isomorphic graphons, then there exist sequences of graphs that are fractionally isomorphic converge and converge to these respective graphons. As an easy but convenient corollary of our methods, we get that every regular graphon can be approximated by regular graphs.
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Permutation Flip Processes
Hladký, Jan ; Řada, Hanka
We introduce a broad class of stochastic processes on permutations which we call flip processes. A single step in these processes is given by a local change on a randomly chosen fixed-sized tuple of the domain. We use the theory of permutons to describe the typical evolution of any such flip process started from any initial permutation. More specifically, we construct trajectories in the space of permutons with the property that if a finite permutation is close to a permuton then for any time it stays with high probability is close to this predicted trajectory. This view allows to study various questions inspired by dynamical systems.
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Structural properties of graphs---probabilistic and deterministic point of view
Hladký, Jan ; Šámal, Robert (advisor) ; Pawlas, Zbyněk (referee)
We study bipartite subgraphs of a random cubic graph in the thesis. We show, that an edge-maximum bipartite subgraph of a random cubic graph on n vertices has asymptotically almost surely less then 3 2 · 0.9351n edges. We also show that the number of vertices of a vertex-maximum induced bipartite subgraph of a random cubic graph lies within interval [0.75n; 0.9082n]. To obtain the lower bound we design a randomized algorithm for finding a large induced bipartite subgraph of a random cubic graph. We discuss consequences of the results for graph homomorphisms, namely for Pentagon Conjecture posed by Nešetřil.
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Structural Graph Theory
Hladký, Jan ; Kráľ, Daniel (advisor) ; Keevash, Peter (referee) ; Krivelevich, Michael (referee)
of doctoral thesis Structural graph theory Jan Hladký In the thesis we make progress on the Loebl-Komlós-Sós Conjecture which is a classic problem in the field of Extremal Graph Theory. We prove the following weaker version of the Conjecture: For every α > 0 there exists a number k0 such that for every k > k0 we have that every n-vertex graph G with at least (1 2 +α)n vertices of degrees at least (1+α)k contains each tree T of order k as a subgraph. The proof of our result follows a strategy common to approaches which employ the Szemerédi Regularity Lemma: the graph G is decomposed, a suitable combinatorial structure inside the decomposition is found, and then the tree T is embedded into G using this structure. However the decomposition given by the Regularity Lemma is not of help when G sparse. To surmount this shortcoming we develop a decomposition technique that applies also to sparse graphs: each graph can be decomposed into vertices of huge degrees, regular pairs (in the sense of the Regularity Lemma), and two other components each exhibiting certain expander-like properties. The results were achieved in a joint work with János Komlós, Diana Piguet, Miklós Simonovits, Maya Jakobine Stein and Endre Szemerédi. 1
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Tomaszewski's conjecture
Toufar, Tomáš ; Šámal, Robert (advisor) ; Hladký, Jan (referee)
In 1986, Boguslaw Tomaszewski asked the following question: Consider n real numbers a1, . . . , an such that the sum of their squares is 1. Of the 2n expressions |ε1a1 + · · · + εnan| with εi = ±1, can there be more with value > 1 than with value ≤ 1? Apart from being of intrinsic interest in probability, an answer to this conjecture would also have applications in quadratic programming. However, even after more than thirty years the conjecture is still unsolved. In this thesis we settle a special case of the conjecture - we prove that the conjecture holds for vectors of the form (α, δ, . . . , δ) of sufficiently large dimension. This generalizes earlier result which showed that the conjecture holds for vectors of the form (δ, . . . , δ). 1
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Structural Graph Theory
Hladký, Jan ; Kráľ, Daniel (advisor) ; Keevash, Peter (referee) ; Krivelevich, Michael (referee)
of doctoral thesis Structural graph theory Jan Hladký In the thesis we make progress on the Loebl-Komlós-Sós Conjecture which is a classic problem in the field of Extremal Graph Theory. We prove the following weaker version of the Conjecture: For every α > 0 there exists a number k0 such that for every k > k0 we have that every n-vertex graph G with at least (1 2 +α)n vertices of degrees at least (1+α)k contains each tree T of order k as a subgraph. The proof of our result follows a strategy common to approaches which employ the Szemerédi Regularity Lemma: the graph G is decomposed, a suitable combinatorial structure inside the decomposition is found, and then the tree T is embedded into G using this structure. However the decomposition given by the Regularity Lemma is not of help when G sparse. To surmount this shortcoming we develop a decomposition technique that applies also to sparse graphs: each graph can be decomposed into vertices of huge degrees, regular pairs (in the sense of the Regularity Lemma), and two other components each exhibiting certain expander-like properties. The results were achieved in a joint work with János Komlós, Diana Piguet, Miklós Simonovits, Maya Jakobine Stein and Endre Szemerédi. 1
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Szemerédi Regularity Lemma a jeho aplikace v kombinatorice
Hladký, Jan ; Kráľ, Daniel (advisor) ; Dvořák, Zdeněk (referee)
In the thesis we provide a solution of the Loebl-Komlós-Sós Conjecture (1995) for dense graphs. We prove that for any q > 0 there exists a number n0 N such that for any n > n0 and k > qn the following holds. Let G be a graph of order n with at least n/2 vertices of degree at least k. Then any tree of order k+1 is a subgraph of G. This improves previous results by Zhao (2002), and Piguet and Stein (2007). A strengthened version of the above theorem together with a lower bound for the problem is discussed. As a corollary a tight bound on the Ramsey number of two trees is stated. The proof of the main theorem combines a Regularity-Lemma based embedding technique with the Stability Method of Simonovits. Results presented here are based on joint work with Diana Piguet.
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Structural properties of graphs---probabilistic and deterministic point of view
Hladký, Jan ; Pawlas, Zbyněk (referee) ; Šámal, Robert (advisor)
We study bipartite subgraphs of a random cubic graph in the thesis. We show, that an edge-maximum bipartite subgraph of a random cubic graph on n vertices has asymptotically almost surely less then 3 2 · 0.9351n edges. We also show that the number of vertices of a vertex-maximum induced bipartite subgraph of a random cubic graph lies within interval [0.75n; 0.9082n]. To obtain the lower bound we design a randomized algorithm for finding a large induced bipartite subgraph of a random cubic graph. We discuss consequences of the results for graph homomorphisms, namely for Pentagon Conjecture posed by Nešetřil.
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