New Prime Vertex Labelings

N. Diefenderfer, M. Hastings, L. Heath, H. Prawzinsky, B. Preston, E. White, & A. Whittemore

Department of Mathematics & Statistics, Northern Arizona University

Simple Graph

Deﬁnition: A simple graph G(V, E) is a set V of vertices and a set E of

edges connecting some (possibly empty) subset of those vertices such that

G(V, E) does not contain loops or multiple edges. The degree of a vertex is

the number of edges “touching” that vertex.

Below is a graph that is not simple.

Families of Graphs

Deﬁnition: For n ≥ 2, a path, denoted Pn

, is a connected graph that

consists of n vertices and n − 1 edges such that 2 vertices have degree 1 and

n − 2 vertices have degree 2.

Deﬁnition: For n ≥ 3, a cycle, denoted Cn

, a connected graph consisting

of n vertices, each of degree 2.

Deﬁnition: A star, denoted Sn

, consists of one vertex of degree n and n

vertices of degree one.

P8

C12

S5

Graph Labeling

Deﬁnition: A graph labeling is an “assignment” of integers (possibly sat-

isfying some conditions) to the vertices, edges, or both.

Deﬁnition: A graph with n vertices has a prime vertex labeling, or is said

to be coprime, 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.

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

.

Cycle Pendant Stars

Deﬁnition: A cycle pendant star is the graph of the form Cn

P2

Sk

and

a generalized cycle pendant star is the graph of the form Cn

P2

Sk

Sk

Sk

· · · Sk

.

Theorem: All cycle pendant stars of the form Cn

P2

Sk

for 3 ≤ k ≤ 8,

and all generalized cycle pendant stars of the form Cn

P2

S3

S3

have a

prime vertex labeling.

Below is a 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

Hairy Cycles

Deﬁnition: An m-hairy cycle is a graph of the form Cn

Sm

.

Theorem: All 3-hairy, 5-hairy and 7-hairy cycles are coprime.

Below is a prime vertex labeling of C4

S7

.

1

2

3

4 5 6

7

8

19

17

18

20

21

22

23

24

11

9

10

12

13

14

15

16

29

25

26

27

28

30

31

32

Generalized Books

Deﬁnition: A book is a graph of the form Sn

× P2

, and a generalized book

is a graph of the form Sn

× Pm

.

Theorem: All Sn

× Pm

are coprime for 3 ≤ m ≤ 7.

Below is a prime vertex labeling of S8

× P6

.

1 2 3 4 5 6

12 11 10 9 8 7

18 17 16 15 14 13

20 21 22 23 24 19

30 29 28 27 26 25

36 35 34 33 32 31

42 41 40 39 38 37

48 47 46 45 44 43

50 51 52 53 54 49

Prisms

Deﬁnition: A prism is a graph of the form Cn

× P2

.

Theorem: If n − 1 is a prime number and n ≥ 4, then Cn

× P2

is coprime.

Below is a prime vertex labeling of C6

× P2

.

5

2

3

4 1

12

6

7

8

9 10

11

Cycle Chains

Deﬁnition: A cycle chain, denoted Cm

n

, is a graph consisting of m distinct

n-cycles symmetrically adjoined by a single vertex on each cycle.

Deﬁnition: A Mersenne prime is a prime number of the form Mn

= 2n −1.

Theorem: All Cm

8

, Cm

6

, Cm

4

and Cm

n

, and where n = 2k and 2k − 1 is a

Mersenne prime have prime vertex labelings.

Here is a prime vertex labeling of 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

Fibonacci Chains

Deﬁnition: A Fibonacci chain, denoted Cm

F

, is a graph consisting of m

cycles adjoined by a single vertex on each cycle, where the ﬁrst cycle contains

3 vertices and the ith cycle contains Fi+1

+ 1 vertices.

Theorem: All Cm

F

are coprime.

Below is an example of a prime vertex labeling of C5

F

.

1

2

4

3 5

6 7

10

9

8

12

11

13

14

15

16

17

18

19

20

21

References

1. Gallian. A dynamic survey of graph labeling. Electron. J. Comb.,

Volume 17, 2014.

2. Diefenderfer, Hastings, Heath, Prawzinsky, Preston, White, Whittemore.

Prime Vertex Labelings of Families of Unicyclic Graphs.

Rose-Hulman Undergraduate Mathematics Journal (to appear).

3. Diefenderfer, Ernst, Hastings, Heath, Prawzinsky, Preston, Rushall,

White, Whittemore. Prime Vertex Labelings of Several Families of

Graphs (submitted).

Directed by Dana C. Ernst & Jeﬀ Rushall Typeset using L

A

TEX, TikZ, and beamerposter