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 Title: Viewing extremal and structural problems through a probabilistic lens Author(s): Delcourt, Michelle Jeannette Director of Research: Balogh, Jozsef Doctoral Committee Chair(s): Kostochka, Alexandr Doctoral Committee Member(s): Kirkpatrick, Kay; Tserunyan, Anush Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Small subgraph conditioning method Random regular graph Intersecting families Star decomposition Structural graph theory Extremal combinatorcs Abstract: This thesis focuses on using techniques from probability to solve problems from extremal and structural combinatorics. The main problem in Chapter 2 is determining the typical structure of $t$-intersecting families in various settings and enumerating such systems. The analogous sparse random versions of our extremal results are also obtained. The proofs follow the same general framework, in each case using a version of the Bollobás Set-Pairs Inequality to bound the number of maximal intersecting families, which then can be combined with known stability theorems. Following this framework from joint work with Balogh, Das, Liu, and Sharifzadeh, similar results for permutations, uniform hypergraphs, and vector spaces are obtained. In 2006, Barát and Thomassen conjectured that the edges of every planar 4-edge-connected 4-regular graph can be decomposed into disjoint copies of $S_3$, the star with three leaves. Shortly afterward, Lai constructed a counterexample to this conjecture. Following joint work with Postle, in Chapter 3 using the Small Subgraph Conditioning Method of Robinson and Wormald, we find that a random 4-regular graph has an $S_3$-decomposition asymptotically almost surely, provided we have the obvious necessary divisibility conditions. In 1988, Thomassen showed that if $G$ is at least $(2k-1)$-edge-connected then $G$ has a spanning, bipartite $k$-connected subgraph. In 1989, Thomassen asked whether a similar phenomenon holds for vertex-connectivity; more precisely: is there an integer-valued function $f(k)$ such that every $f(k)$-connected graph admits a spanning, bipartite $k$-connected subgraph? In Chapter 4, as in joint work with Ferber, we show that every $10^{10}k^3 \log n$-connected graph admits a spanning, bipartite $k$-connected subgraph. Issue Date: 2017-03-27 Type: Thesis URI: http://hdl.handle.net/2142/97669 Rights Information: Copyright 2017 Michelle Delcourt Date Available in IDEALS: 2017-08-10 Date Deposited: 2017-05
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