Dana Ernst
March 20, 2015
150

# Nonunicyclic Graphs with Prime Vertex Labelings, II

This talk was given by my undergraduate research student Alyssa Whittemore on March 20, 2015 at the 2015 MAA/CURM Spring Conference at Brigham Young University.

This research was supported by the National Science Foundation grant #DMS-1148695 through the Center for Undergraduate Research (CURM).

March 20, 2015

## Transcript

1. ### 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
2. ### 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.

4. ### 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.
5. ### 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.
6. ### 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 :

3 ≤ m ≤ 7.
8. ### 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 ← →
9. ### 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 →
10. ### 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 ← →
11. ### 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 ← ← →
12. ### 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....
13. ### 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.
14. ### 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.

16. ### Acknowledgments Center for Undergraduate Research in Mathematics Northern Arizona University

Research Advisors Dana Ernst and Jeff Rushall
17. ### 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
18. ### 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