Dana Ernst
January 24, 2015
110

Prime Vertex Labelings of Graphs

This poster was presented by my undergraduate research students Hannah Prawzinsky and Emily White 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

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