Dana Ernst
March 20, 2015
130

# Nonunicyclic Graphs with Prime Vertex Labelings, II

This talk was given by my undergraduate research student Alyssa Whittemore 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, II
Alyssa Whittemore
Joint work with: Nathan Diefenderfer, Michael Hastings, Levi
Heath, Hannah Prawzinsky, Briahna Preston & Emily White
CURM 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.

3. Inﬁnite Families of Graphs
P8
C12
S5

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. 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.

6. 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
:

7. Book Generalizations
Theorem
All Sn × Pm
are prime for 3 ≤ m ≤ 7.

8. Example of Sn × P3
1 2 3
6 5 4
8 9 7
12 11 10
14 15 13
18 17 16
20 21 19 A

A

A

9. Example of Sn × P4
1 2 3 4
6 7 8 5
12 11 10 9
16 15 14 13
18 19 20 17
24 23 22 21
28 27 26 25 ←

A

A

10. Example of Sn × P5
1 2 3 4 5
10 9 8 7 6
12 13 14 15 11
18 19 20 17 16
24 23 22 25 21
30 29 28 27 26
32 33 34 35 31
38 39 40 37 36 B
A

C
B
A

11. Example of Sn × P6
1 2 3 4 5 6
12 11 10 9 8 7
18 17 16 15 14 13
20 21 22 23 24 19
30 29 28 27 26 25
36 35 34 33 32 31
42 41 40 39 38 37
48 47 46 45 44 43
50 51 52 53 54 49 A

A

12. Summary of Results
• Sn × P3
has a 2-row repeating pattern: ←, A.
• Sn × P4
has a 3-row repeating pattern: A, ←, ←.
• Sn × P5
has a 6-row repeating pattern: A, B, C, ←, A, B.
• Sn × P6
has a 5-row repeating pattern: ←, ←, ←, ←, A.
The simplicity ends here....

13. Sn × P7
We have found a prime vertex labeling for Sn × P7
.
There are 10 ordered row permutations. A pattern of ordered
row permutations begins at row 3 and repeats in blocks of 30 for
as many rows as needed.

14. Conjecture
Conjecture
Sn × Pm
is prime for all m ≥ 2 and n ≥ 1.
Here is an example of Sn × P12
, speciﬁcally 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
We have found a prime labeling that works for more than
205,626 rows (2,467,524 vertices) of this family.

15. Future Work
Larger Generalized Books?

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

17. Book Generalizations
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

18. 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