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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