#### Date of Award

Spring 5-1-2023

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### First Advisor

Guantao Chen

#### Second Advisor

Florian Enescu

#### Third Advisor

Hendricus Van der Holst

#### Fourth Advisor

Zhongshan Li

#### Fifth Advisor

Yi Zhao

#### Abstract

Let $G$ be a graph, $V (G)$ and $E(G)$ be the vertex set and edge set of $G$, respectively. A perfect matching of $G$ is a set of edges, $M\subseteq E(G)$, such that each vertex in $G$ is incident with exactly one edge in $M$. An $r$-regular graph is said to be an $r$-graph if $|\partial(X)| \geq r$ for each odd set $X \subseteq V(G)$, where $|\partial(X)|$ denotes the set of edges with precisely one end in $X$. One of the most famous conjectures in Matching Theory, due to Berge, states that every 3-graph $G$ has five perfect matchings such that each edge of $G$ is contained in at least one of them. Likewise, generalization of the Berge Conjecture given, by Seymour, asserts that every $r$-graph $G$ has $2r-1$ perfect matchings that covers each $e \in E(G)$ at least once. In the first part of this thesis, I will provide a lower bound to number of perfect matchings needed to cover the edge set of an $r$-graph. I will also present some new conjectures that might shade a light towards the generalized Berge conjecture. In the second part, I will present a proof of a conjecture stating that there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that mean subtree order of $H$ is smaller then mean subtree order of $G$.

#### DOI

https://doi.org/10.57709/35334148

#### Recommended Citation

Tokar, Nizamettin, "Berge - Fulkerson Conjecture And Mean Subtree Order." Dissertation, Georgia State University, 2023.

doi: https://doi.org/10.57709/35334148

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