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# New Results on Prime Vertex Labelings, I

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

3. 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.

4. Connected and Unicyclic 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.
Deﬁnition
A unicyclic graph is a simple graph containing exactly one
cycle.
Here is a unicyclic graph that is NOT connected.

5. Pendants
Deﬁnition
In a unicyclic graph, a pendant is a path on two vertices with
exactly one vertex being a cycle vertex. The non-cycle vertex
of a pendant is called a pendant vertex.

6. Inﬁnite Families of Graphs
P8
C12
S5

7. Gluing Function
Deﬁnition
G1
G2
is the graph that results from ”selectively gluing” copies
of G2
to some vertices of G1
.
Here is C3
P2
S3
.

8. 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

9. Prime Vertex Labeling
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.

10. Conjecture
Conjecture (Seoud and Youssef, 1999)
All unicyclic graphs have a prime vertex labeling.

11. Known Prime Vertex Labelings
1 2 3 4 5 6 7 8
P8
1
12
11
10
9
8
7
6
5
4
3
2
C12
1
2
6
5
4 3
S5

12. New Inﬁnite Families of Graphs
Deﬁnition
A cycle pendant star is a cycle with each cycle vertex
adjacent to an identical star, denoted Cn
P2
Sk
.
Deﬁnition
A generalized cycle pendant star is a cycle with each cycle
vertex adjacent to an identical star and each noncentral star
vertex is a central vertex for an identical star, denoted
Cn
P2
Sk
Sk
Sk · · · Sk
.

13. Example of C4
P2
S4

14. Prime Vertex Labelings of Inﬁnite Families of
Graphs
Theorem
For all n ∈ N, we have:
The cycle pendant stars, Cn
P2
Sk
are prime for 3 ≤ k ≤ 8.
The generalized cycle pendant star, Cn
P2
S3
S3
has a
prime vertex labeling.
The cases for k ≥ 9 for cycle pendant stars and all other cases
of generalized cycle pendant stars remain open.

15. Labeling Function of Cn
P2
S3
Labeling Function
Let c1, c2, . . . , cn
denote the cycle labels, p1, p2, . . . , pn
denote
the vertices adjacent to the corresponding cycle vertices and
the pendant vertices adjacent to 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

16. Prime Vertex Labeling of Cn
P2
S3
1
3
4
2 5
11
13
14
12
15
6
7
9
8
10

17. Generalized Labeling of Cn
P2
S4
6i − 5 6i − 1
6i − 4
6i − 3
6i − 2
6i
6i − 11
6i + 1

18. Prime Vertex Labeling of C4
P2
S4
1
5
2
3
4
6
19
23
20
21
22
24
7
11
8
9
10
12
13
17
14
15
16
18

19. Prime Vertex Labeling of C5
P2
S6
1
5
2
3 4 6 7
8
9
13
10
11
12
14
15
16
17
19
18
20
21
22
23
24
25
29
26
27
28
30
31
32
33
37
34
35
36
38
39
40

20. EXCELing with Computer Aid

21. Prime Vertex Labeling of C4
P2
S7
1
5
2
3
4
6
7
8
9
10
13
11
12
14
15
16
17
18
19
23
20
21
22
24
25
26
27
28
31
29
30
32
33
34
35
36

22. Prime Vertex Labeling of Cn
P2
S3
S3
1
2
5
9
11
3 4
6
7
8
10
12
13
14
15
16
19
23
25
17
18
20
21
22
24
26
27
28
29
32
31
35
41
30
33
34
36
37
38
39
40
42
43
44
47
51
53
45
46
48
49
50
52
54
55 56

23. Future Work
Conjecture
For all n, k ∈ N, Cn
P2
Sk
is prime.
Conjecture
For all n, k ∈ N, Cn
P2
Sk
Sk
Sk · · · Sk
is prime.

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

25. Labeling Function of Cn
P2
S3
S3
Note that Cn
P2
S3
S3
contains 14n vertices. We will identify
our vertices as follows. Let ci, 1 ≤ i ≤ n denote the cycle
vertices, let pi
denote the pendant vertex adjacent to ci
, let
the non-cycle vertices adjacent to pi
be denoted si,j
for
1 ≤ j ≤ 3, and let the remaining vertices adjacent to each si,j
be denoted li,j,k
for 1 ≤ k ≤ 3. Our labeling function
f : V → {1, 2, . . . , 14n} is best deﬁned by ﬁrst describing cycle
and pendent vertex labels:
f(ci) = 14i − 13, 1 ≤ i ≤ n
f(pi) =
14i − 12, i ≡3
1, 2
14i − 10, i ≡3
0

26. Labeling Function of Cn
P2
S3
S3
The remaining vertex labels are determined by the values of i, j
and k as follows. If i ≡3
1, 2, then deﬁne
f(si,j) =

14i − 9, j = 1
14i − 5, j = 2
14i − 3, j = 3
f(li,j,k ) =

14i − 11, j = 1, k = 1
14i − 10, j = 1, k = 2
14i − 8, j = 1, k = 3
14i − 7, j = 2, k = 1
14i − 6, j = 2, k = 2
14i − 4, j = 2, k = 3
14i − 2, j = 3, k = 1
14i − 1, j = 3, k = 2
14i, j = 3, k = 3

27. Labeling Function of Cn
P2
S3
S3
If i ≡3
0, then deﬁne
f(si,j) =

14i − 11, j = 1
14i − 7, j = 2
14i − 1, j = 3
f(li,j,k ) =

14i − 12, j = 1, k = 1
14i − 9, j = 1, k = 2
14i − 8, j = 1, k = 3
14i − 6, j = 2, k = 1
14i − 5, j = 2, k = 2
14i − 4, j = 2, k = 3
14i − 3, j = 3, k = 1
14i − 2, j = 3, k = 2
14i, j = 3, k = 3

28. Labeling Function of Cn
P2
S4
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

29. Labeling Function of Cn
P2
S5
Let ci
, 1 ≤ i ≤ n, denote the cycle vertices, let pi
denote the
, and let oik
, 1 ≤ k ≤ 5, denote
the outer vertices adjacent to pi
. The labeling function
f : V → {1, 2, . . . 7n} is given by:
f(ci) = 7i − 6, 1 ≤ i ≤ n
f(pi) =

7i − 2, i ≡6
1, 3
7i − 3, i ≡6
2, 4
7i − 4, i ≡6
5
7i − 5, i ≡6
0, i ≡30
0
7i − 1, i ≡30
0

30. Labeling Function of Cn
P2
S5
f(oi1) =
7i − 5, i ≡6
0 or i ≡30
0
7i − 4, i ≡6
0, i ≡30
0
f(oi2) =
7i − 4, i ≡6
0, 5 or i ≡30
0
7i − 3, i ≡6
5 or i ≡6
0, i ≡30
0
f(oi3) =
7i − 3, i ≡6
0, 2, 4, 5 or i ≡30
0
7i − 2, i ≡6
1, 3
f(oi4) =
7i − 2, i ≡30
0
7i − 1, i ≡30
0
f(oi5) = 7i, 1 ≤ i ≤ n

31. Labeling Function of Cn
P2
S6
Let ci
, 1 ≤ i ≤ n, denote the cycle vertices, let pi
denote the
, and let oik
, 1 ≤ k ≤ 6, denote
the outer vertices adjacent to pi
. The labeling function
f : V → {1, 2, . . . 8n} is given by:
f(ci) = 8i − 7, 1 ≤ i ≤ n
f(pi) =

8i − 3, i ≡3
0
8i − 5, i ≡3
0, i ≡15
0
8i − 1, i ≡15
0

32. Labeling Function of Cn
P2
S6
f(oi1) = 8i − 6, 1 ≤ i ≤ n
f(oi2) =
8i − 5, i ≡3
0 or i ≡15
0
8i − 3, i ≡3
0, i ≡15
0
f(oi3) = 8i − 4, 1 ≤ i ≤ n
f(oi4) =
8i − 3, i ≡15
0
8i − 1, i ≡15
0
f(oi5) = 8i − 2, 1 ≤ i ≤ n
f(oi6) = 8i, 1 ≤ i ≤ n

33. Labeling Function of Cn
P2
S7
Let ci
, 1 ≤ i ≤ n, denote the cycle vertices, let pi
denote the
, and let oik
, 1 ≤ k ≤ 7, denote
the outer vertices adjacent to pi
. The labeling function
f : V → {1, 2, . . . 9n} is given by:
f(ci) = 9i − 8, 1 ≤ i ≤ n
f(pi) =

9i − 4, i ≡2
1
9i − 5, i ≡2
0, i ≡10
0
9i − 7i ≡10
0, i ≡70
0
9i − 1, i ≡70
0

34. Labeling Function of Cn
P2
S7
f(oi1) =
9i − 7, i ≡10
0 or i ≡70
0
9i − 5, i ≡10
0, i ≡70
0
f(oi2) = 9i − 6, 1 ≤ i ≤ n
f(oi3) =
9i − 5, i ≡2
1 or i ≡70
0
9i − 4, i ≡2
0, i ≡70
0
f(oi4) =
9i − 4, i ≡70
0
9i − 1, i ≡70
0
f(oi5) = 9i − 3, 1 ≤ i ≤ n
f(oi6) = 9i − 2, 1 ≤ i ≤ n
f(oi7) = 9i, 1 ≤ i ≤ n

35. Labeling Function of Cn
P2
S8
Let ci
, 1 ≤ i ≤ n, denote the cycle vertices, let pi
denote the
, and let oik
, 1 ≤ k ≤ 8, denote
the outer vertices adjacent to pi
. The labeling function
f : V → {1, 2, . . . 10n} is given by:
f(ci) = 10i − 9, 1 ≤ i ≤ n
f(pi) =

10i − 3, i ≡3
0
10i − 7, i ≡3
0, i ≡21
0
10i − 1, i ≡21
0

36. Labeling Function of Cn
P2
S8
f(oi1) = 10i − 8, 1 ≤ i ≤ n
f(oi2) =
10i − 7, i ≡21
0 or i ≡3
1, 2
10i − 3, i ≡3
0, i ≡21
0
f(oi3) = 10i − 6, 1 ≤ i ≤ n
f(oi4) = 10i − 5, 1 ≤ i ≤ n
f(oi5) = 10i − 4, 1 ≤ i ≤ n
f(oi6) =
10i − 3, i ≡21
0
10i − 1, i ≡21
0
f(oi7) = 10i − 2, 1 ≤ i ≤ n
f(oi8) = 10i, 1 ≤ i ≤ n