Dana Ernst
January 24, 2015
190

# Prime Vertex Labelings of Graphs

This talk was given by my undergraduate research students Briahna Preston and Alyssa Whittemore on January 25, 2015 at the Nebraska Conference for Undergraduate Women in Mathematics at the University of Nebraska-Lincoln.

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

January 24, 2015

## Transcript

1. Prime Vertex Labelings of Graphs
Briahna Preston & Alyssa Whittemore (joint work with Nathan
Diefenderfer, Michael Hastings, Levi Heath, Hannah
Prawzinsky, & Emily White)
NCUWM
January 24, 2015
Funded by a CURM Mini-grant

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. Examples of Graphs
Deﬁnition
A path, denoted Pn
, consists of n vertices and n − 1 edges such
that 2 vertices have degree 1 and n − 2 vertices have degree 2.
Deﬁnition
A cycle, denoted Cn
, is a closed “loop” with n vertices and n
edges with every vertex of degree 2.
Deﬁnition
A star, denoted Sn
, consists of one vertex of degree n and n
vertices of degree one.

4. Constructing New Graphs From Old Graphs
Deﬁnition
G1
G2
is the graph that results from “selectively gluing” copies
of G2
to some vertices of G1
.
Here is C3
P2
S3
.

5. Prime Vertex Labelings
Deﬁnition
A graph with n vertices has a prime vertex labeling if its vertices
are labeled with the integers 1, 2, 3, . . . , n such that no two
vertices have the same label and every pair of 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 Labeling
Use integers 1–12
relatively prime labels
No two vertices can have
identical labels

6. Families with Prime Vertex Labelings
1
2
3
4
5 6
1
11
9
7
5 3
2
12
10
8
6 4
Among the inﬁnite families known to have prime vertex
labelings are stars, pendant graphs, cycles, paths, ladders,

7. Families without Prime Vertex Labelings
3
2
6 5
4
1
Among the inﬁnite families known to not have prime vertex
labelings are Wn
if n odd, Kn
if n ≥ 4, K2 + Cn
if n ≥ 4, ...

8. Our Research Project
Our research project involves ﬁnding new inﬁnite families of
graphs with prime vertex labelings. Part of our focus has been
on the following conjecture:
Conjecture (Seoud, 1990)
All unicyclic graphs (graphs containing exactly one cycle)
have a prime vertex labeling.

9. New Results
One of the simplest types of unicyclic graphs is a tadpole,
which is a cycle with one path attached. It is trivial to show that
all tadpoles have a prime vertex labeling.
It is somewhat less trivial to prove the following:
Theorem 1 (Preston & Whittemore)
All double- and triple-tailed tadpoles have prime vertex
labelings.
9 2
3
4
5
6
7
8
10
11
12
13
1 17
16
15
14

10. New Results
Theorem 2 (Preston & Whittemore et al.)
The following inﬁnite families of unicyclic graphs have prime
vertex labelings:
Cn
P2
S3
Cn
P2
S3
S3
Cn
P2
Sk
for 3 ≤ k ≤ 8
Cn
S3
Cn
S5
Cn
S7

11. New Results
Here is an example for Cn
S5
, speciﬁcally C4
S5
:
1
4
5
6
2
3
23
21
22
24
19
20
17
15
16
18
13
14
11
9
10
12
7
8

12. New Results
Theorem 3 (Preston & Whittemore et al.)
The following inﬁnite families of graphs (non-unicyclic) have
prime vertex labelings:
Sn × P4
Sn × P6
Cn × P2
if n + 1 is a prime number

13. New Results
Here is an example of the prime labeling for Sn × P6
, speciﬁcally
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

14. New Results
Theorem 4 (Preston & Whittemore et al.)
The family Cn × Pi
if n is odd for all i > 1 does not have a prime
vertex labeling.
1
5
4 3
2
7
6
10 9
8

15. Future Work
Conjectures
Here are some inﬁnite families we hope to show have prime
vertex labelings:
Cn
P2
Sk
for all n ≥ 3, k ≥ 3
Sn × P12
All unicyclic graphs (Seoud’s Conjecture)

16. Acknowledgements
Jeff Rushall and Dana Ernst, advisors
Department of Mathematics & Statistics at Northern Arizona
University
The Center for Undergraduate Research in Mathematics
Mathematics Organizers

17. QUESTIONS?

18. Proof
An odd-cycle double-tailed tadpole with n vertices is labeled
with the following labeling function:
f(vi) =

i + 1, for i ≤ k
i + 2, for k < i ≤ n − 2
k + 1, for i = n
1, for i = n − 1
Note that v1
must be the vertex adjacent to an intersection
point on the even arc length of the cycle and vk
is the other
intersection point.

19. Ternary Trees
Deﬁnition
A complete ternary tree is a directed rooted tree with every
internal vertex having 3 children.
Here are examples of 1 and 2-level complete ternary trees:
a
b
c
d a
b
c
d
b1
b2
b3
c1 c2
c3
d1
d2
d3

20. New Results
Theorem 1
Every Cn
P2
S4
has a prime vertex labeling.
Theorem 2
Every Cn
P2
S4
S4
has a prime vertex labeling.

21. Theorem 1 Labeling
Let c1, c2, . . . , cn
denote the cycle labels, let p1, p2, . . . , pn
denote the vertices adjacent to the corresponding cycle
vertices and let the pendant vertices off of pi
be denoted
si,j, 1 ≤ j ≤ 3. The labeling function f : V → {1, 2, . . . , 5n} is given
by:
f(ci) = 5i − 4, 1 ≤ i ≤ n
f(pi) =
5i − 2, if i is odd
5i − 3, if i is even
f(si,j) =

5i − 3 + j, i is even
5i − 2 + j, j = 3 and i is odd
5i − 3, j = 3 and i is odd

22. Example of Theorem 1 Labeling
1
3
4
2 5
11
13
14
12
15
6
7
9
8
10

23. Cycle Pendant Stars
Deﬁnition
A star is a graph with one vertex adjacent to all other vertices,
each of which has degree one.
Deﬁnition
A cycle pendant star is a cycle with each cycle vertex
adjacent to an identical star, denoted Cn
P2
Sk
.

24. Example C4
P2
S5

25. Cn
P2
S5
Theorem
Cn
P2
S5
is prime for all n.
Labeling
Let ci
, 1 ≤ i ≤ n, denote the cycle vertices, let pi
denote the
, and let oik
, 1 ≤ k ≤ 4, denote
the outer vertices adjacent to pi
. The labeling function
f : V → {1, 2, . . . 6n} is given by:
f(ci) = 6i − 5, 1 ≤ i ≤ n
f(pi) = 6i − 1, 1 ≤ i ≤ n
f(oi1) = 6i − 2, 1 ≤ i ≤ n
f(oi2) = 6i − 3, 1 ≤ i ≤ n
f(oi3) = 6i − 4, 1 ≤ i ≤ n
f(oi4) = 6i, 1 ≤ i ≤ n

26. General Labeling
6i − 5 6i − 1
6i − 2
6i − 3
6i − 4
6i
6i − 11
6i + 1

27. Labeling Example C4
P2
S5
1
5
2
3
4
6
19
23
20
21
22
24
7
11
8
9
10
12
13
17
14
15
16
18

28. Cycle Pendant Trees
Theorem
Cn
P2
Sk
is prime for all n ∈ N and 4 ≤ k ≤ 9.
Conjecture
Cn
P2
Sk
is prime for all n, k ∈ N.

29. Hairy Cycles
An m-hairy cycle is a cycle Cn
with m paths of length 2
connected to each vertex of the cycle, denoted Cn
mP2
.
We have found a prime labeling for 2, 3, 5, and 7-hairy
cycles.
Here is an example of a 2-hairy cycle, denoted C3
2P2
.
a
b c
d
e
f
h
i
j

30. Cycles with 3 Hairs
Theorem
Cn
3P2
has a prime labeling.
Labeling
Let c1, c2, . . . cn
denote the vertices of Cn
, and let the pendant
be denoted pi
j
, 1 ≤ j ≤ 3. The labeling
function f : V → {1, 2, . . . 4n} is given by:
f(ci) =
1 i = 1
4i − 1 i ≥ 2
f(pi
j
) =

j + 1 i = 1, 1 ≤ j ≤ 3
4i − 3 i ≥ 2, j = 1
4i − 2 i ≥ 2, j = 2
4i i ≥ 2, j = 3

31. Example of a Cycle with Three Hairs
1
2
3
4
15
13
14
16
7
5
6
8
11
9
10
12

32. 5-Hairy Cycle
Theorem
Cn
5P2
has a prime labeling for all n.
Labeling
Let c1, c2, . . . , cn
denote the vertices of Cn
, and let the pendant
be denoted pi
j
, 1 ≤ j ≤ 5. The labeling
formula f : V → {1, 2, . . . , 6n} is given by:
f(ci) =
1, i = 1
6(i − 1) + 5, i ≥ 2
f(pi
j
) =

j + 1, 1 ≤ j ≤ 5, i = 1
6(i − 1) + j, 1 ≤ j ≤ 4, i ≥ 2
6(i − 1) + 6, j = 5, i ≥ 2

33. Example of 5-Hairy Cycle
1
4
5
6
2
3
23
21
22
24
19
20
17
15
16
18
13
14
11
9
10
12
7
8

34. Cycles with 7 Hairs
Theorem
Cn
7P2
has a prime labeling
Labeling
Let c1, c2, . . . cn
denote the vertices of Cn
, and let the pendant
be denoted pi
j
, 1 ≤ j ≤ 7. The labeling
formula f : V → {1, 2, . . . 8n} is given by:
f(c1) = 1
f(p1
j
) = j + 1
f(ci) =

8i − 5 i ≡ 2, 3, 6, 8, 9, 11, 12, 14 (mod 15)
8i − 3 i ≡ 1, 4, 5, 7, 10, 13 (mod 15)
8i − 1 i ≡ 0 (mod 15)
f(pi
j
) = {8i − 7, 8i − 6, . . . , 8i} \ {f(ci)}

35. Example of a Cycle with 7 Hairs
1
2
3
4 5 6
7
8
19
17
18
20
21
22
23
24
11
9
10
12
13
14
15
16

36. Shotgun Weed Graph
Bertrand’s Postulate
For every n > 1, there exists a prime p such that n ≤ p ≤ 2n.
We can build a graph that is prime using this fact.
1
2
13
10
9
11
14
7
12
8
5
4
3
6

37. And Now For Something Completely Different...
Theorem
If n is odd then Cn × Pi
is not prime for all i > 1.
Proof.
Parity argument.
1
5
4 3
2
7
6
10 9
8

38. Hastings Helms
Theorem
If n + 1 is prime then Cn × P2
is prime.
Labeling
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

39. Example of C6 × P2
(Hastings Helms)
1
2 3
4
5
6
8
9 10
11
12
7

40. 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.

41. 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
:

42. Results
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

43. Results
Theorem
All Sn × P6
are prime.

44. Results
Labeling
Let f : V → 1, 2, . . . , 6(n + 1) denote our labeling function. Let
c1, c2, c3, c4, c5, c6
form a path through the center of each Sn
.
Let
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, 23 (mod30). Then
let vi,n
be the vertex in the ith star and in the nth row. Then
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(p) = 5
f(vi,k ) = vi,k−1 + 6 for 1 ≤ i ≤ 6 and 2 ≤ k ≤ n

45. Results
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

46. 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

47. Future Work
Hairier Hairy Cycles?
Larger Generalized Books?
Larger Complete Ternary Trees?
Lollipop Graphs?
Unrooted Complete Binary Trees?
ALL Unicyclic Graphs?????????

48. Acknowledgements
Jeff Rushall and Dana Ernst, advisors
Department of Mathematics & Statistics at Northern Arizona
University
The Center for Undergraduate Research in Mathematics

49. QUESTIONS?