New Results on Prime Vertex Labelings, III Michael Hastings, Hannah Prawzinsky & Alyssa Whittemore Joint work with: Nathan Diefenderfer, Levi Heath, Briahna Preston & Emily White SUnMaRC February 28, 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

Hastings Helms Definition A Hastings Helm, 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.

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

Hastings Helms 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 (Hastings Helms) 1 2 3 4 5 6 8 9 10 11 12 7

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

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

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

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 × P4 are prime. 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 Theorem All Sn × P6 are prime.

Book Generalizations Labeling Let vi,n be the vertex in the ith star and the nth row, and let c1, c2, c3, c4, c5, c6 form a path through the center of each Sn . Then our labeling function f : V → {1, 2, . . . , 6(n + 1)} is given by: f(c1) = 6, f(c2) = 1, f(c3) = 2, f(c4) = 3, f(c5) = 4, and f(c6) = p, the largest prime ≡ 1, 7, 13, 17, or 23 (mod 30). Note that p = 6t + s where s = 1, 5. Let f(vs,t ) = 5. Finally, f(v1,1) = 7 f(v2,1) = 8 f(v3,1) = 9 f(v4,1) = 10 f(v5,1) = 11 f(v6,1) = 12 f(vi,k ) = vi,k−1 + 6 for 1 ≤ i ≤ 6 and 2 ≤ k ≤ n

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

Conjecture Conjecture Sn × P12 is prime. We have found a prime labeling that works for more than 205,626 rows (2,467,524 vertices). Here is an example of 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

Future Work Larger Generalized Books? Other Hastings Helms? Cycle chains with larger or odd cycles?

Acknowledgments SUnMaRC Organizers Center for Undergraduate Research in Mathematics Northern Arizona University Office of Undergraduate Research, NAU Research Advisors Dana Ernst and Jeff Rushall