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# 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.

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. 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.

6. Example of C4
8

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

23. Book Generalizations
Theorem
All Sn × P6
are prime.

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