#### Title of Abstract

Prime Labeling Graphs and Hypergraphs

#### Poster Number

071

#### College

College of Arts and Sciences

#### Department

Department of Mathematics

#### Faculty Mentor

Arran Hamm, Ph.D

#### Abstract

Graph labeling problems date back to the beginning of Graph Theory itself (see the Four Color Theorem). Roughly 40 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., each edge’s labels have greatest common divisor one). In the 1980s, Entriger conjectured that a certain family of graphs all have prime labelings; our work furthered the progress on this conjecture by giving a prime labeling for several members of this family. Additionally, we studied graph parameters related to the coprime graph. The coprime graph on *n* vertices is the graph whose vertices are numbered 1, 2, … , *n* with i~j if and only if i and j are coprime. Using the graph parameters we calculated, we were able to conclude that several classes of graphs are not prime. We concluded our work by examining this notion generalized to hypergraphs (which allow “edges” to have size larger than two) and give a class of hypergraphs which are not prime.

#### Previously Presented/Performed?

Mathematical Association of America, Southeastern Section Meeting, Clemson University, March 2018

#### Grant Support?

Supported by an SC INBRE grant from the National Institute of General Medical Sciences (NIH-NIGMS)

#### Start Date

20-4-2018 2:15 PM

#### End Date

20-4-2018 4:15 PM

Prime Labeling Graphs and Hypergraphs

Richardson Ballroom (DIGS)

Graph labeling problems date back to the beginning of Graph Theory itself (see the Four Color Theorem). Roughly 40 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., each edge’s labels have greatest common divisor one). In the 1980s, Entriger conjectured that a certain family of graphs all have prime labelings; our work furthered the progress on this conjecture by giving a prime labeling for several members of this family. Additionally, we studied graph parameters related to the coprime graph. The coprime graph on *n* vertices is the graph whose vertices are numbered 1, 2, … , *n* with i~j if and only if i and j are coprime. Using the graph parameters we calculated, we were able to conclude that several classes of graphs are not prime. We concluded our work by examining this notion generalized to hypergraphs (which allow “edges” to have size larger than two) and give a class of hypergraphs which are not prime.