Electronic Theses and Dissertations Archive

Date

2026

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

Alexander Gavrilyuk

Committee Chair

Dheer Desai

Committee Member

Ben McCarty

Committee Member

David Grynkiewicz

Committee Member

Nathan Lindzey

Abstract

Erdös-Ko-Rado (EKR) type theorems carry the answer to the following fundamental optimization question in extremal combinatorics, given a finite family of combinatorial objects alongside a notion of intersection, what is the maximum size of a pairwise intersecting subfamily? In classical EKR type problems, the maximum subfamily is obtained by fixing one common core element or set and taking all members of the family containing it. In this dissertation, we show new results in EKR theory for families of edge intersecting spanning trees of complete bipartite graphs. We provide sharp asymptotic upper bounds for the maximum size of these t-edge intersecting families. As opposed to classical EKR theorems where trivial t-intersecting families are extremal, maximal families of t-intersecting spanning trees of K_(n,n) vary in the following cases: when t = 1, they consist of all trees containing a fixed edge together with all stars, bi-stars, and almost bi-stars that do not contain that edge. When t = 2, they contain a 2-matching, and a set of size at most 2n^n + n^4-2n^3, consisting of all bi-stars of K_(n,n), along with all stars and almost bi-stars having precisely one common edge and no (almost) saturated vertex in the 2-matching. When 3 ≤t≤n/(4224 log_2⁡n ) , they involve of the set of all spanning trees of K_(n,n) containing a t-matching F of K_(n,n), and a set of at most 2n^n + n^4-2n^3 stars, all of which contain a (t-1)-matching in F and their saturated vertices are not in F. These results are based on joint work with Gordian Bruns, Dheer Noal Desai, Alexander Gavrilyuk, and Nathan Lindzey. We utilize a recently developed technique called spread approximation. The core idea is to show that any large family of t-intersecting spanning trees of K_(n,n) consists of all spanning trees having a fixed set of t disjoint edges, together with a relatively small collection of exceptional trees. Moreover, to show that no other spanning trees can appear in a maximum size family, we use a spectral argument involving Laplacian eigenvalue estimates via Weyl’s inequalities and Kirchhoff’s matrix-tree theorem to count the number of spanning trees in the t-star family avoiding a given spanning tree outside the t-matching. Thus, a maximum size family cannot contain any spanning trees beyond the non-exceptional and t-star family members, because otherwise the size of the t-intersecting family is not optimal. These exceptional trees are completely characterized, they are precisely the stars, bi-stars, and almost bi-stars of K_(n,n). Extending these results to unbalanced complete bipartite host graphs is a natural subsequent phase, as well as investigating the setting where the host graphs are complete multipartite graphs.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest/Clarivate.

Notes

Open Access

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