Nonunicyclic Graphs with Prime Vertex Labelings, II Alyssa Whittemore Joint work with: Nathan Diefenderfer, Michael Hastings, Levi Heath, Hannah Prawzinsky, Briahna Preston & Emily White CURM Conference March 20, 2015

What is a Graph? Definition A graph G(V, E) is a set V of vertices and a set E of edges connecting some (possibly empty) subset of those vertices. A simple graph is a graph that contains neither “loops” nor multiple edges between vertices. A connected graph is a graph in which there exists a “path” between every pair of vertices. For the remainder of the presentation, all graphs are assumed to be simple and connected.

Infinite Families of Graphs P8 C12 S5

Prime Vertex Labelings Definition An n-vertex graph has a prime vertex labeling if its vertices are labeled with the integers 1, 2, 3, . . . , n such that no label is repeated and all adjacent vertices (i.e., vertices that share an edge) have labels that are relatively prime. 1 6 7 4 9 2 3 10 11 12 5 8 Some useful number theory facts: • All pairs of consecutive integers are relatively prime. • Consecutive odd integers are relatively prime. • A common divisor of two integers is also a divisor of their difference. • The integer 1 is relatively prime to all integers.

Books Definition A book is the graph Sn × P2 , where Sn is the star with n pendant vertices and P2 is the path with 2 vertices. Here is a picture of S4 × P2 : 3 5 7 9 4 6 8 10 2 1 It is known that all books have a prime labeling.

Book Generalizations Definition A generalized book is a graph of the form Sn × Pm , which looks like m − 1 books glued together. Here is a picture of S5 × P4 :

Book Generalizations Theorem All Sn × Pm are prime for 3 ≤ m ≤ 7.

Example of Sn × P3 1 2 3 6 5 4 8 9 7 12 11 10 14 15 13 18 17 16 20 21 19 A ← A ← A ← →

Example of Sn × P4 1 2 3 4 6 7 8 5 12 11 10 9 16 15 14 13 18 19 20 17 24 23 22 21 28 27 26 25 ← ← A ← ← A →

Example of Sn × P5 1 2 3 4 5 10 9 8 7 6 12 13 14 15 11 18 19 20 17 16 24 23 22 25 21 30 29 28 27 26 32 33 34 35 31 38 39 40 37 36 B A ← C B A ← →

Example of Sn × P6 1 2 3 4 5 6 12 11 10 9 8 7 18 17 16 15 14 13 20 21 22 23 24 19 30 29 28 27 26 25 36 35 34 33 32 31 42 41 40 39 38 37 48 47 46 45 44 43 50 51 52 53 54 49 A ← ← ← ← A ← ← →

Summary of Results • Sn × P3 has a 2-row repeating pattern: ←, A. • Sn × P4 has a 3-row repeating pattern: A, ←, ←. • Sn × P5 has a 6-row repeating pattern: A, B, C, ←, A, B. • Sn × P6 has a 5-row repeating pattern: ←, ←, ←, ←, A. The simplicity ends here....

Sn × P7 We have found a prime vertex labeling for Sn × P7 . There are 10 ordered row permutations. A pattern of ordered row permutations begins at row 3 and repeats in blocks of 30 for as many rows as needed.

Conjecture Conjecture Sn × Pm is prime for all m ≥ 2 and n ≥ 1. Here is an example of Sn × P12 , specifically S3 × P12 : 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 10 33 34 35 36 37 38 39 40 11 42 5 44 45 46 7 48 2 3 4 1 6 43 8 9 32 47 12 41 We have found a prime labeling that works for more than 205,626 rows (2,467,524 vertices) of this family.

Future Work Larger Generalized Books?

Acknowledgments Center for Undergraduate Research in Mathematics Northern Arizona University Research Advisors Dana Ernst and Jeff Rushall

Book Generalizations Here is an example of S5 × P4 : 24 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 3 23 2 1 4

Book Generalizations Here is an example of the prime labeling for Sn × P6 , in particular, S4 × P6 : 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 5 24 25 26 27 28 29 30 6 1 2 3 4 23