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# New Prime Vertex Labelings

This poster was presented by my undergraduate research students Nathan Diefenderfer, Michael Hastings, Levi Heath, Hannah Prawzinsky, Briahna Preston, Emily White, and Alyssa Whittemore (Northern Arizona University) on April 25, 2015 at the 2015 NAU Undergraduate Symposium at Northern Arizona University in Flagstaff, AZ.

This research was supported by the National Science Foundation grant #DMS-1148695 through the Center for Undergraduate Research (CURM). April 24, 2015

## Transcript

1. 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