#### Event Title

On Prime Labelings of Berge Hypergraphs of Nonprime Graphs

#### Session Title

STEM and Biomedical Research

#### Document Type

Oral Presentation

#### College

College of Arts and Sciences

#### Department

Department of Mathematics

#### Honors Thesis Committee

Arran Hamm, Ph.D.

#### Description

Graph labeling problems, such as the Four Color Conjecture, date back to the beginning of Graph Theory itself. Roughly forty years ago, the notion of a prime labeling of a graph was introduced: a graph on *n* vertices has a *prime labeling* if its vertices can be labeled by the numbers 1, 2, …, *n*, so that each edge spans a coprime pair (i.e. the labels on an edge have greatest common divisor one). A *hypergraph* consists of “edges” on a vertex set where the edges may contain any number of vertices. Since greatest common divisor can be generalized to more than two numbers, it is natural to consider prime labeling hypergraphs. As an entry point to this problem, we focus on a subclass of hypergraphs referred to as *Berge hypergraphs*. Given a graph *G*, the hypergraph is Berge-*G* if there is a matching between edges of *G* and edges of the hypergraph in which, in this matching, each edge is within the corresponding edge of the hypergraph. The paper gives a condition based on how “close” *G* is to being prime, which implies that any hypergraph which is Berge-*G* is prime, and finds that a handful of *G* for which any hypergraph which is Berge-*G* is prime.

On Prime Labelings of Berge Hypergraphs of Nonprime Graphs

Graph labeling problems, such as the Four Color Conjecture, date back to the beginning of Graph Theory itself. Roughly forty years ago, the notion of a prime labeling of a graph was introduced: a graph on *n* vertices has a *prime labeling* if its vertices can be labeled by the numbers 1, 2, …, *n*, so that each edge spans a coprime pair (i.e. the labels on an edge have greatest common divisor one). A *hypergraph* consists of “edges” on a vertex set where the edges may contain any number of vertices. Since greatest common divisor can be generalized to more than two numbers, it is natural to consider prime labeling hypergraphs. As an entry point to this problem, we focus on a subclass of hypergraphs referred to as *Berge hypergraphs*. Given a graph *G*, the hypergraph is Berge-*G* if there is a matching between edges of *G* and edges of the hypergraph in which, in this matching, each edge is within the corresponding edge of the hypergraph. The paper gives a condition based on how “close” *G* is to being prime, which implies that any hypergraph which is Berge-*G* is prime, and finds that a handful of *G* for which any hypergraph which is Berge-*G* is prime.