# Nonunicyclic Graphs with Prime Vertex Labelings, I

This talk was given by my undergraduate research students Michael Hastings & Hannah Prawzinsky 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 Michael Hastings & Hannah

Prawzinsky Joint work with: Nathan Diefenderfer, Levi Heath, Briahna Preston, Emily White & Alyssa Whittemore 2015 MAA/CURM Spring 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. ### Cycle Chains Deﬁnition A cycle chain, denoted Cm n ,

is a graph that consists of m different n-cycles adjoined by a single vertex on each cycle (each cycle shares a vertex with its adjacent cycle(s)). Here we show labelings for Cm 4 , Cm 6 , and Cm 8 . The labelings for these three inﬁnite families of graphs all employ similar strategies.

7. ### Cycle Chain Results Theorem All Cm 8 are prime using

the labeling function: f(ci,k ) = 7i + k − 6 Theorem All Cm 6 are prime using the labeling function: f(ci,k ) = 5i + k − 4 Theorem All Cm 4 are prime using the labeling function f(ci,k ) = 3i + k − 1
8. ### Labeled C5 8 1 2 3 4 5 6 7

8 15 11 10 9 1 12 13 14 19 18 17 16 15 22 21 20 29 25 24 23 19 26 27 28 33 32 31 30 29 36 35 34
9. ### Labeled C5 6 1 2 3 4 5 6 11

8 7 1 9 10 16 13 12 11 14 15 19 18 17 16 21 20 26 23 22 19 24 25
10. ### Labeled C4 4 5 4 3 2 7 6 5

8 11 9 7 10 13 12 11 1
11. ### Labeled C5 4 5 4 3 2 7 6 5

8 11 9 7 10 13 12 11 14 1 15 13 16
12. ### Mersenne Primes Deﬁnition A Mersenne prime is a prime number

of the form Mn = 2n − 1. There are 48 known Mersenne primes. The ﬁrst few Mersenne primes are: M2 = 22 − 1 = 3 M3 = 23 − 1 = 7 M5 = 25 − 1 = 31
13. ### Theorem All Cm n , where n = 2k and

2k − 1 is a Mersenne prime, have prime labelings.
14. ### Fibonacci Chains Fibonacci sequence The sequence, {Fn}, of Fibonacci numbers

is deﬁned by the recurrence relation Fn = Fn−1 + Fn−2 , where F1 = 1 and F2 = 1. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . Proposition Any two consecutive Fibonacci numbers in the Fibonacci sequence are relatively prime. Theorem Fibonacci Chains, denoted Cn F , are prime for all n ∈ N where n is the number of cycles that make up the Fibonacci chain.
15. ### Fibonacci Chains (C5 F ) 1 2 4 3 5

6 7 10 9 8 12 11 13 14 15 16 17 18 19 20 21
16. ### Prisms Deﬁnition A Prism, denoted Hn , is the graph

Cn × P2 . Here we will show that if n is odd, then Hn is not prime and will show that if either n + 1 or n − 1 is prime, then Hn has a prime vertex labeling. The remaining cases are currently open. We conjecture that Hn is prime for all even n.
17. ### Prisms Theorem If n is odd, then Hn is not

prime. Proof. Parity argument. 1 5 4 3 2 7 6 10 9 8
18. ### Prisms Theorem If n + 1 is prime, then Hn

is prime. Labeling function Let c1 1 , c1 2 , . . . , c1 n denote the vertices on the inner cycle, and c2 1 , c2 2 , ..., c2 n be the vertices on the outer cycle. The labeling formula f : V → {1, 2, . . . , 2n} is given by: f(cj i ) =      i, i = 1, 2, . . . , n, j = 1 i + n + 1, i = 1, 2, . . . , n − 1, j = 2 n + 1, i = n, j = 2
19. ### Example of H6 (Prisms) 1 2 3 4 5 6

8 9 10 11 12 7
20. ### Prisms Theorem If n − 1 is prime, then Hn

is prime. Labeling function Let c1 1 , c2 1 , . . . , cn 1 denote the vertices on the ”inner” cycle and c1 2 , c2 2 , . . . , cn 2 denote the corresponding vertices on the ”outer” cycle. The labeling formula f : V → {1, 2, . . . , 2n} is given by: f(ci 1 ) =          i, i = 2, 3, . . . n − 2 n − 1, i = 1 1, i = n − 1 2n, i = n f(ci 2 ) = i + n − 1, i = 2, 3, . . . n n, i = 1
21. ### Example of H8 (Prisms) 5 4 3 2 1 16

7 6 12 11 10 9 8 15 14 13
22. ### Example of H8 (Prisms) 5 4 3 2 7 16

1 6 12 11 10 9 8 15 14 13
23. ### Example of H32 (Prisms) 17 16 15 14 13 12

11 10 9 8 7 6 5 4 3 2 31 64 1 30 29 28 27 26 25 24 23 22 21 20 19 18 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49
24. ### General labeling for Hn when n − 1 is prime

n − 1 2 n − 2 1 2n n n + 1 2n − 3 2n − 2 2n − 1
25. ### Future Work Cycle chains with larger or odd cycles? Other

cases for Prisms? Generalized Prisms?
26. ### Acknowledgments Center for Undergraduate Research in Mathematics Northern Arizona University

Research Advisors Dana Ernst and Jeff Rushall