New Coprime Vertex Labelings Hannah Prawzinsky Joint work with: Nathan Diefenderfer, Michael Hastings, Levi Heath, Briahna Preston, Emily White & Alyssa Whittemore NCUWM January 30, 2016

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.

Simple graphs Definition A simple graph is a graph that contains neither “loops” nor multiple edges between vertices. For the remainder of the presentation, all graphs are assumed to be simple. Here is a graph that is NOT simple.

Connected and unicyclic graphs Definition 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 connected. Definition A unicyclic graph is a simple graph containing exactly one cycle. Here is a unicyclic graph that is NOT connected.

Infinite families of graphs P8 C12 S5

Graph labelings Definition A graph labeling is an “assignment” of integers (possibly satisfying some conditions) to the vertices, edges, or both. Formal graph labelings are functions. 2 3 2 3 1 4 1 4 1 2 3 4 1 2 3 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.

Original motivation for research Conjecture (Seoud and Youssef, 1999) All unicyclic graphs have a prime vertex labeling. Though our research lead to many new results for unicyclic graphs, some of which were presented last year, this talk will primarily focus on a a specific family of non-unicyclic graphs.

Known prime labelings 1 2 3 4 5 6 7 8 P8 1 12 11 10 9 8 7 6 5 4 3 2 C12 1 2 6 5 4 3 S5

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 , Cm 6 , Cm 4 have prime vertex labelings.

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

Mersenne cycle chains 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

Future Work and Acknowledgements Future Work • Seoud and Youssef’s Conjecture Acknowledgments • NCUWM Organizers • University of Nebraska—Lincoln • Center for Undergraduate Research in Mathematics • Northern Arizona University • Research Advisors Dana Ernst and Jeff Rushall

Questions ?

A Labeled 7-Hairy Cycle 1 2 3 4 5 6 7 8 19 17 18 20 21 22 23 24 11 9 10 12 13 14 15 16 29 25 26 27 28 30 31 32

Prime Vertex Labeling of C5 P2 S6 1 5 2 3 4 6 7 8 9 13 10 11 12 14 15 16 17 19 18 20 21 22 23 24 25 29 26 27 28 30 31 32 33 37 34 35 36 38 39 40

Prime Vertex Labeling of Cn P2 S3 S3 1 2 5 9 11 3 4 6 7 8 10 12 13 14 15 16 19 23 25 17 18 20 21 22 24 26 27 28 29 32 31 35 41 30 33 34 36 37 38 39 40 42 43 44 47 51 53 45 46 48 49 50 52 54 55 56

A Labeled Bertrand Weed Graph 1 2 13 10 9 11 14 7 12 8 5 4 3 6

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

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