Dissertation/ Thesis
Clique Factors: Extremal and Probabilistic Perspectives
| Τίτλος: | Clique Factors: Extremal and Probabilistic Perspectives |
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| Συγγραφείς: | Morris, Patrick |
| Συνεισφορές: | male, Szabó, Tibor, Morris, Robert |
| Έτος έκδοσης: | 2022 |
| Συλλογή: | FU Berlin: Refubium |
| Θεματικοί όροι: | Probabilistic Combinatorics, Extremal Combinatorics, Pseudorandom Graphs, Random Graphs, Spanning structures in graphs, Graph Theory, Clique Factors, ddc:510 |
| Περιγραφή: | A K_r-factor in a graph G is a collection of vertex-disjoint copies of K_r covering the vertex set of G. In this thesis, we investigate these fundamental objects in three settings that lie at the intersection of extremal and probabilistic combinatorics. Firstly, we explore pseudorandom graphs. An n-vertex graph is said to be (p,β)-bijumbled if for any vertex sets A, B ⊆ V (G), we have e( A, B) = p| A||B| ± β√|A||B|. We prove that for any 3 ≤ r ∈ N and c > 0 there exists an ε > 0 such that any n-vertex (p, β)-bijumbled graph with n ∈ rN, δ(G) ≥ c p n and β ≤ ε p^{r −1} n, contains a K_r -factor. This implies a corresponding result for the stronger pseudorandom notion of (n, d, λ)-graphs. For the case of K_3-factors, this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result, we can conclude that the same condition of β = o( p^2n) actually guarantees that a (p, β)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2. Secondly, we explore the notion of robustness for K_3-factors. For a graph G and p ∈ [0, 1], we denote by G_p the random sparsification of G obtained by keeping each edge of G independently, with probability p. We show that there exists a C > 0 such that if p ≥ C (log n)^{1/3}n^{−2/3} and G is an n-vertex graph with n ∈ 3N and δ(G) ≥ 2n/3 , then with high probability G_p contains a K_3-factor. Both the minimum degree condition and the probability condition, up to the choice of C, are tight. Our result can be viewed as a common strengthening of the classical extremal theorem of Corrádi and Hajnal, corresponding to p = 1 in our result, and the famous probabilistic theorem of Johansson, Kahn and Vu establishing the threshold for the appearance of K_3-factors (and indeed all K_r -factors) in G (n, p), corresponding to G = K_n in our result. It also implies a first lower bound on the number of K_3-factors ... |
| Τύπος εγγράφου: | doctoral or postdoctoral thesis |
| Περιγραφή αρχείου: | xvii, 236 Seiten; application/pdf |
| Γλώσσα: | English |
| DOI: | 10.17169/refubium-34642 |
| Διαθεσιμότητα: | https://refubium.fu-berlin.de/handle/fub188/34924 https://doi.org/10.17169/refubium-34642 https://nbn-resolving.org/urn:nbn:de:kobv:188-refubium-34924-5 |
| Rights: | http://www.fu-berlin.de/sites/refubium/rechtliches/Nutzungsbedingungen |
| Αριθμός Καταχώρησης: | edsbas.8B69D977 |
| Βάση Δεδομένων: | BASE |
| DOI: | 10.17169/refubium-34642 |
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