Date of Award

Summer 2025

Document Type

Open Access Thesis

Department

Mathematics

First Advisor

Ruth Luo

Abstract

An $n$-vertex graph $G$ is \textit{hamiltonian} if it contains a length $n$ cycle as a subgraph. A stronger notion of hamiltoncity is pancyclicity. A graph $G$ is \textit{pancyclic} if $G$ contains a cycle of length $k$ for every $3\leq k \leq n$. Dirac's classical result on hamiltonicity states that if an $n$-vertex graph $G$ has minimum degree $\delta(G)\geq \frac{n}{2}$, then $G$ is hamiltonian. Under the same minimum degree condition as Dirac, J. A. Bondy showed that $G$ must also be pancyclic or the exceptional graph $K_{\frac{n}{2}, \frac{n}{2}}$. In 1971, Bondy proposed a so called meta-conjecture claiming that \textit{``almost any nontrivial condition on a graph which implies that the graph is hamiltonian also implies that the graph is pancyclic,”} which he later amended to include that \textit{``...there may be a simple family of exceptional graphs."} An $n$-vertex hypergraph $\cH = (V,E)$ is defined by a vertex set and hyperedge set. A hypergraph $\cH$ is $r$-uniform if $|e| = r$ for every $e\in E$. A Berge cycle of length $k$ is an alternating sequence of $k$ distinct vertices and $k$ distinct edges $v_0e_0v_1e_1\dots v_{k-1}e_{k-1}v_0$ such that $v_i,v_{i+1} \subseteq e_i$ for all $i$, with indices taken modulo $k$. The definitions of Berge hamiltonicity and Berge pancyclicity are analogous to those for graphs, with the exception that we consider cycles of length 2 for Berge pancyclicity. This thesis considers an analog of Bondy's meta conjecture for $r$-uniform hypergraphs. After a discussion on Bondy's meta-conjecture and known results on Berge cycles in hypergraphs, we show that for $r$-uniform hypergraphs the same conditions which guarantee Berge hamiltoncity, also guarantee Berge pancyclicity for sufficiently large $n$, which is shown to be sharp with respect to the minimum degree condition.

Rights

© 2025, Teegan Cole Bailey

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