Prime Labeling Graphs and Hypergraphs
Poster Number
071
College
College of Arts and Sciences
Department
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.
