Graduation Year

2025

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Brendan Nagle, Ph.D.

Co-Major Professor

Theodore Molla, Ph.D.

Committee Member

Brian Curtin, Ph.D.

Committee Member

Jarred Ligatti, Ph.D.

Keywords

Bipartite Hypergraphs, Regularity Lemma, Fano Plane

Abstract

Of a given bipartite graph G = (V, E), it is elementary to construct a bipartition in timeO(|V |+|E|). For a given k-graph H = H(k) with k ≥ 3 fixed, Lovász proved that deciding whether H is bipartite is NP-complete. In this thesis, we consider the average running time of this problem. For that, let B_n = (B^(k))_n be the collection (family) of all bipartite k-graphs H on the fixed vertex set [n] = {1, ...., n}. We construct, of a given H ∈ B_n, a bipartition in time averaging O(n^k) over the class B_n. We provide two proofs of our result. When k = 3, this result expedites one of Person and Schacht, which it also extends to all k ≥ 3. In this thesis, we also consider another related result of Person and Schacht cast in the setting of k = 3. Let Fn be the family of [n]-vertex 3-graphs containing no copies of the Fano plane as a subhypergraph. Since the Fano plane is not bipartite, it holds that B_n ⊆ F_n, which Person and Schacht proved is nearly an equality (in a sense made precise later). They also established an algorithm which properly and minimally colors a given H ∈ F_n in time averaging O(n^5 (log n)^2) over the class F_n. We expedite this average time for the closely related concept of an optimal 2-partition [n] = X_H ∪˙ Y_H of a fixed H ∈ F_n, which minimizes over all 2-partitions [n] = X ∪˙ Y the number e_H(X) + e_H(Y ) of edges e ∈ E_H with e ⊆ X or e ⊆ Y We give an algorithm which constructs, of a given H ∈ F_n, an optimal 2-partition [n] = X_H ∪˙ Y_H of H in time averaging O(n^3) over the class F_n.

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