Dana Ernst
February 28, 2015
150

# New Results on Prime Vertex Labelings, III

This talk was given by my undergraduate research students Michael Hastings, Hannah Prawzinsky, and Alyssa Whittemore on February 28, 2015 at the at the Southwestern Undergraduate Mathematics Research Conference (SUnMaRC) at the University of Texas at El Paso.

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

## Dana Ernst

February 28, 2015

## Transcript

1. ### 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
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. ### 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.
13. ### 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
14. ### 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
15. ### Example of H6 (Hastings Helms) 1 2 3 4 5

6 8 9 10 11 12 7
16. ### 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
17. ### Example of H8 (Hastings Helms) 5 4 3 2 1

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

16 1 6 12 11 10 9 8 15 14 13
19. ### 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
20. ### 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.
21. ### 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 :
22. ### 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

24. ### 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
25. ### 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
26. ### 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
27. ### Future Work Larger Generalized Books? Other Hastings Helms? Cycle chains

with larger or odd cycles?
28. ### Acknowledgments SUnMaRC Organizers Center for Undergraduate Research in Mathematics Northern

Arizona University Ofﬁce of Undergraduate Research, NAU Research Advisors Dana Ernst and Jeff Rushall