Full Record

Author | Delcourt, Michelle Jeannette |

Title | Viewing extremal and structural problems through a probabilistic lens |

URL | http://hdl.handle.net/2142/97669 |

Publication Date | 2017 |

Date Accessioned | 2017-08-10 20:32:42 |

Degree | PhD |

Discipline/Department | Mathematics |

Degree Level | doctoral |

University/Publisher | University of Illinois – Urbana-Champaign |

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. |

Subjects/Keywords | Small subgraph conditioning method; Random regular graph; Intersecting families; Star decomposition; Structural graph theory; Extremal combinatorcs |

Contributors | Balogh, Jozsef (advisor); Kostochka, Alexandr (Committee Chair); Kirkpatrick, Kay (committee member); Tserunyan, Anush (committee member) |

Language | en |

Rights | Copyright 2017 Michelle Delcourt |

Country of Publication | us |

Record ID | handle:2142/97669 |

Repository | uiuc |

Date Indexed | 2020-03-09 |

Grantor | University of Illinois at Urbana-Champaign |

Issued Date | 2017-03-27 00:00:00 |

Sample Search Hits | Sample Images | Cited Works

…further direction; the same can be
said about certain structural problems.
Star Decompositions of *Random* *Regular* Graphs
As Barát and Thomassen [11] note, decompositions of the edges of a graph G into copies of a small fixed
subgraph can be…

…translate this structural problem to
the setting of *random* d-*regular* graphs:
Theorem 3.1.8.
A *random* 4-*regular* graph on n vertices has an orientation with out-degrees 0 or 3
asymptotically almost surely, provided that 2n is divisible by 3.
Although this…

…appears to be a straightforward application of the second moment method [4], standard
probabilistic techniques do not work here; if Y = Y (n) is the number of orientations of a *random* 4-*regular*
q
2
]
3
graph on n vertices with out…

…families of subgraphs).
A recent line of investigation is extending classical results to the so-called “sparse *random* setting”. If
fn,m (H) is the number of labeled H-free graphs on n labeled vertices with precisely m edges, then the…

…also enumerate such systems and explore the sparse *random* setting. As demonstrated in this thesis for
permutations and vector spaces, this work fits into a more general framework that could be adapted for a
variety of other settings provided the…

…trivial, and there are nt t! + o(1) 2(n−t)! t-intersecting families.
Additionally, we prove two results in the sparse *random* setting. First we obtain the following sparse
extension of Theorem 2.3.4. Let (Sn )p denote the p…

…*random* subset of Sn , where each permutation in
3
Sn is included independently with probability p. Provided p is not too small, we show that with high
probability the largest t-intersecting family in (Sn )p is trivial. Note that the work by…

…n, p) denote the p-*random* k-uniform hypergraph on [n], in which every edge in
[n]
k
is included
independently at *random* with probability p. Balogh, Bohman, and Mubayi [7] initiated the study of
intersecting…