Dana Ernst
March 20, 2015
160

# Nonunicyclic Graphs with Prime Vertex Labelings, I

This talk was given by my undergraduate research students Michael Hastings & Hannah Prawzinsky 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
Michael Hastings & Hannah Prawzinsky
Joint work with: Nathan Diefenderfer, Levi Heath, Briahna
Preston, Emily White & Alyssa Whittemore
2015 MAA/CURM Spring 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. 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. 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

13. Theorem
All Cm
n
, where n = 2k and 2k − 1 is a Mersenne prime, have
prime labelings.

14. 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.

15. 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

16. Prisms
Deﬁnition
A Prism, 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.

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

18. Prisms
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

19. Example of H6
(Prisms)
1
2 3
4
5
6
8
9 10
11
12
7

20. Prisms
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

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

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

23. Example of H32
(Prisms)
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
31
64
1
30
29
28
27
26 25 24
23
22
21
20
19
18
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
63
62
61
60
59
58
57 56 55
54
53
52
51
50
49

24. 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

25. Future Work
Cycle chains with larger or odd cycles?
Other cases for Prisms?
Generalized Prisms?

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