Dana Ernst
December 05, 2014
160

Prime Vertex Labelings of Graphs

This talk was given by my undergraduate research students Nathan Diefenderfer, Michael Hastings, Levi Heath, Hannah Prawzinsky, Briahna Preston, Emily White, and Alyssa Whittemore on December 5, 2014 at the Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) at Northern Arizona University.

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

Dana Ernst

December 05, 2014

Transcript

1. Prime Vertex Labelings of Graphs
Nathan Diefenderfer, Michael Hastings, Levi Heath, Hannah
Prawzinsky, Briahna Preston, Emily White, & Alyssa
Whittemore
FAMUS
December 5, 2014

2. Why are we here?
CURM (Center for Undergraduate Research in Math)
Mini-grant
MAT 485
Conferences
NCUWM - 1/24 (Lincoln, NE)
SUnMaRC - 2/27 (El Paso, TX)
CURM Conference - 3/20 (Provo, UT)

3. 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.

4. Simple Graphs
Deﬁnition
A simple graph is a graph that contains neither “loops” nor
multiple edges between vertices.
For the rest of the presentation, all graphs are assumed to be
simple. Here is a graph that is NOT simple.

5. Connected Graphs
Deﬁnition
A connected graph is a graph in which there exists a “path”
between every pair of vertices.
For the rest of the presentation, all graphs are assumed to be
connected.
Here is a graph that is NOT connected.

6. Cycles
Deﬁnition
A cycle is a connected path consisting of a single “circuit,” and
hence has equal numbers of vertices and edges. We let Cn
denote an n-vertex cycle.
Below is an image of C12
.

7. Unicyclic Graphs
Deﬁnition
A unicyclic graph (simple and connected) is a graph
containing exactly one cycle.
Deﬁnition
G1
G2
is the graph that results from “selectively gluing” copies
of G2
to some vertices of G1
.

8. Example
Here is C3
S5
.

9. Graph Labelings
Deﬁnition
A graph labeling is an “assignment” of integers (possibly
satisfying some conditions) to the vertices, edges, or both.
Formal graph labelings, as you will soon see, are functions.
2 3 2 3
1 4 1 4
1
2
3
4
1
2 3
4

10. 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 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
Adjacent vertices must have
relatively prime labels
No two vertices can have
identical labels

11. Conjecture
Conjecture (Seoud)
All unicyclic graphs have a prime vertex labeling.

12. Cycles and Prime Vertex Labelings
Theorem
All cycles have a prime vertex labeling.
1
2
3
4
5
6
7
8
9
10
11
12

13. More Examples
Here are examples of inﬁnite families of graphs with prime
vertex labelings.
1 2 3 4 5 6 7 8
16 15 14 13 12 11 10 9
1 2 3 4 5 6 7 8

14. More Examples (continued)
1
2
3
4
5 6
1
11
9
7
5 3
2
12
10
8
6 4
The following presenters will discuss some of their original
ﬁndings for speciﬁc families of graphs.

15. 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

16. 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.

17. 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

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

19. 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
.

20. Example C4
P2
S5

21. 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
pendant vertex adjacent to ci
, 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

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

23. 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

24. 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.

25. 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

26. 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
vertices adjacent to ci
be denoted pj
i
, 1 ≤ j ≤ 3. The labeling
function f : V → {1, 2, . . . 4n} is given by:
f(ci) =
1 i = 1
4i − 1 i ≥ 2
f(pj
i
) =

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

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

28. 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 pj
i
, 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(pj
i
) =

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

29. 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

30. 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
vertices adjacent to ci
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)}

31. 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

32. 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

33. 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

34. 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

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

36. 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.

37. 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
:

38. 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

39. Results
Theorem
All Sn × P6
are prime.

40. 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

41. 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

42. 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

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

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

45. QUESTIONS?