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

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.

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.

Example of C4 8

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

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

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

Labeled C4 4 5 4 3 2 7 6 5 8 11 9 7 10 13 12 11 1

Labeled C5 4 5 4 3 2 7 6 5 8 11 9 7 10 13 12 11 14 1 15 13 16

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

Theorem All Cm n , where n = 2k and 2k − 1 is a Mersenne prime, have prime labelings.

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.

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

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.

Prisms Theorem If n is odd, then Hn is not prime. Proof. Parity argument. 1 5 4 3 2 7 6 10 9 8

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

Example of H6 (Prisms) 1 2 3 4 5 6 8 9 10 11 12 7

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

Example of H8 (Prisms) 5 4 3 2 1 16 7 6 12 11 10 9 8 15 14 13

Example of H8 (Prisms) 5 4 3 2 7 16 1 6 12 11 10 9 8 15 14 13

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

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

Future Work Cycle chains with larger or odd cycles? Other cases for Prisms? Generalized Prisms?

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