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On Prime Vertex Labelings

Hannah Prawzinsky & Emily White, Directed by Dana C. Ernst & Jeﬀ Rushall

Department of Mathematics & Statistics, Northern Arizona University

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.

Examples

Deﬁnition

A simple graph is a graph that contains neither “loops” nor multiple edges

between vertices.

Example

Here is an example of a graph that is not simple.

Deﬁnition

1. The path Pn

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

have degree 1 and n − 2 vertices have degree 2. Here is P8

.

2. The cycle Cn

consists of n vertices and n edges such that each vertex has

degree 2. Here is C12

.

3. The star Sn

consists of one vertex of degree n and n vertices of degree one.

Here is S5

.

Deﬁnition

G1 ⋆ G2

is the graph that results from “selectively gluing” copies of G2

to

speciﬁed vertices of G1

. Here is C4 ⋆ P2 ⋆ S4

.

Deﬁnition

A graph labeling is an “assignment” of integers (possibly satisfying some con-

ditions) to the vertices, edges, or both. Formal graph labelings are functions.

Here is a vertex labeling, an edge labeling, and a vertex-edge labeling, respec-

tively, for C4

.

2 3 2 3

1 4 1 4

1

2

3

4

1

2 3

4

Deﬁnition

A graph with n vertices has a prime vertex labeling if its vertices can be labeled

with the integers 1, 2, 3, . . . , n such that every pair of adjacent vertices (i.e.,

vertices that share an edge) have labels that are relatively prime.

Theorem

1. All paths have a prime vertex labeling.

1 2 3 4 5 6 7 8

2. All cycles have a prime vertex labeling.

1

2

3

4

5

6

7

8

9

10

11

12

3. All starts have a prime vertex labeling.

1

2

3

4

5 6

Conjecture (Seoud)

All unicyclic graphs (i.e., graphs containing exactly one cycle) have a prime vertex

labeling.

Theorem (Prawzinksky & White et al.)

1. Every Cn ⋆ P2 ⋆ S3

has a prime vertex labeling.

2. Every Cn ⋆ P2 ⋆ S3 ⋆ S3

has a prime vertex labeling.

Labeling for Part 1 of Theorem

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

Example

Here is an example of the labeling for part 1 of our theorem.

1

3

4

2 5

11

13

14

12

15

6

7

9

8

10

Additional New Results

We have also discovered prime vertex labelings for the following graphs:

1. Double and triple-tailed tadpoles

2. Cn ⋆ P2 ⋆ Sk

for 3 ≤ k ≤ 8

3. Cn ⋆ S3

, Cn ⋆ S5

, Cn ⋆ S7

4. Cn ⋆ 3P2

, Cn ⋆ 5P2

, Cn ⋆ 7P2

In addition, we have obtained results for a few families of non-unicyclic graphs.

Email: Hannah [[email protected]], Emily [[email protected]] Typeset using L

A

TEX, TikZ, and beamerposter