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Sufficient conditions for embedding trees
Rozhoň, Václav ; Klimošová, Tereza (advisor) ; Dvořák, Zdeněk (referee)
We study sufficient degree conditions that force a host graph to contain a given class of trees. This setting involves some well-known problems from the area of extremal graph theory. The most famous one is the Erdős-Sós conjecture that asserts that every graph with average degree greater than k − 1 contains any tree on k + 1 vertices. Our two main results are the following. We prove an approximate version of the Erdős-Sós conjecture for dense graphs and trees with sublinear max- imum degree. We also study a natural refinement of the Loebl-Komlós-Sós conjecture and prove it is approximately true for dense graphs. Both results are based on the so-called regularity method. The second mentioned result is a joint work with T. Klimošová and D. Piguet. 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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Sufficient conditions for embedding trees
Rozhoň, Václav ; Klimošová, Tereza (advisor) ; Dvořák, Zdeněk (referee)
We study sufficient degree conditions that force a host graph to contain a given class of trees. This setting involves some well-known problems from the area of extremal graph theory. The most famous one is the Erdős-Sós conjecture that asserts that every graph with average degree greater than k − 1 contains any tree on k + 1 vertices. Our two main results are the following. We prove an approximate version of the Erdős-Sós conjecture for dense graphs and trees with sublinear max- imum degree. We also study a natural refinement of the Loebl-Komlós-Sós conjecture and prove it is approximately true for dense graphs. Both results are based on the so-called regularity method. The second mentioned result is a joint work with T. Klimošová and D. Piguet. 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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