Dana Ernst
March 20, 2015
220

# Unicyclic Graphs with Prime Vertex Labelings, II

This talk was given by my undergraduate research students Nathan Diefenderfer and Briahna Preston on March 20, 2015 at the 2015 MAA/CURM Spring Conference at Brigham Young University.

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

March 20, 2015

## Transcript

1. Unicyclic Graphs with Prime Vertex
Labelings, II
Nathan Diefenderfer and Briahna Preston
Joint work with: Michael Hastings, Levi Heath, Hannah
Prawzinsky, Emily White & Alyssa Whittemore
2015 MAA/CURM Spring Conference
March 20, 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. A
simple graph is a graph that contains neither “loops” nor
multiple edges between vertices. A connected graph is a
graph in which there exists a “path” between every pair of
vertices.
For the remainder of the presentation, all graphs are assumed
to be simple and connected.

3. Graph Theory Terminology
Deﬁnition
Two vertices are considered adjacent if there is an edge
between them.
Deﬁnition
The degree of a vertex is the number of edges having that
vertex as an endpoint.
Deﬁnition
A subgraph H of a graph G is a graph whose vertex set is a
subset of that of G, and whose adjacency relation is a subset of
that of G restricted to this subset.

4. Inﬁnite Families of Graphs
P8
C12
S5

5. Prime Vertex Labelings
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.

6. 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

7. Unicyclic Graphs
Deﬁnition
A graph is unicyclic if it contains only one cycle as a subgraph.
Conjecture (Seoud and Youssef, 1999)
All unicyclic graphs have a prime labeling.
5
1
2
3 4
9
8 10
7
6

Deﬁnition
A tadpole is a unicyclic graph composed of a cycle and a
single path starting at what we call an intersection point, a
vertex in the cycle.
Deﬁnition
An arc length is the number of cycle edges between two
intersection points. So an even or odd arc length refers to the
parity of the number of edges in the arc length.

Theorem
All double-tailed tadpoles have a prime labeling.
Labeling - Case 1
The most simple case, a double-tailed tadpole with n vertices
and one intersection point vk
, is labeled with the following
labeling function.
f(vi) =

i + 1, for i < k
1, for i = k
i, for i > k

10. A Labeled 6-Cycle Double-Tailed Tadpole
3
4
5
6
1 2
13
12
11
7
8
9
10

Labeling - Case 2
An even-cycle double-tailed tadpole with n vertices and two
distinct intersection points is labeled with the following labeling
function:
f(vi) =
i + 1, for i < n
1, for i = n
Note that v1
must be the vertex adjacent to an intersection
point.

12. A Labeled 8-Cycle Double-Tailed Tadpole
4
5
6
7
8
9
2
3
1 15 14 13
10
11
12

Labeling - Case 3
An odd-cycle double-tailed tadpole with n vertices and two
distinct intersection points is labeled with the following labeling
function:
f(vi) =

i + 1, for i ≤ k
i + 2, for k < i ≤ n − 2
k + 1, for i = n
1, for i = n − 1
Note that v1
must be the vertex adjacent to an intersection
point on the even arc length of the cycle and vk
is the other
intersection point.

14. A Labeled 7-Cycle Double-Tailed Tadpole
5
6
7
8
9
2
3
4 1 14
10
11
12
13

Theorem
All odd-cycle triple-tailed tadpoles have a prime labeling.
Labeling - Case 1
An odd-cycle triple tailed tadpole with n vertices, three distinct
intersection points, and three odd tails is labeled with the
following function:
f(vi) =

i + 1, for 1 ≤ i ≤ 2
i + 2, for 3 ≤ i < n − 2
4, for i = j
1, for i = k

16. A Labeled 9-cycle Triple-Tailed Tadpole
7
8
9
10
11
2
3
5
6
1 20 19
12
13
14
15
16
4 17 18

17. Hairy Cycles
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.
Deﬁnition
An m-hairy cycle, denoted Cn
Sm
is a cycle with m pendants

18. Basic Approach
For an m-hairy cycle:
• N is split into sets of m + 1 consecutive integers.
• Each set is assigned to a cycle vertex and its pendants.
• A number relatively prime to the rest is assigned to the
cycle vertex.
• Odd m values were used so the number of evens and odds
in each set was consistent.

19. 3-Hairy Cycle
Theorem
For all n ≥ 3, Cn
S3
has a prime labeling.
Labeling
Let c1, c2, . . . cn
denote the vertices of Cn
, and let the pendant
be denoted pj
i
, 1 ≤ j ≤ 3. The labeling
function f : V → {1, 2, . . . 4n} is given by:
f(ci) =
1 i = 1
4i − 1 i ≥ 2
f(pj
i
) =

j + 1 i = 1, 1 ≤ j ≤ 3
4i − 3 i ≥ 2, j = 1
4i − 2 i ≥ 2, j = 2
4i i ≥ 2, j = 3

20. A Labeled 3-Hairy Cycle
1
2
3
4
15
13
14
16
7
5
6
8
11
9
10
12

21. 5-Hairy Cycle
Theorem
For all n ≥ 3, Cn
S5
has a prime labeling.
Labeling
Let c1, c2, . . . , cn
denote the vertices of Cn
, and let the pendant
be denoted pj
i
, 1 ≤ j ≤ 5. The labeling
formula f : V → {1, 2, . . . , 6n} is given by:
f(ci) =
1, i = 1
6(i − 1) + 5, i ≥ 2
f(pj
i
) =

j + 1, 1 ≤ j ≤ 5, i = 1
6(i − 1) + j, 1 ≤ j ≤ 4, i ≥ 2
6(i − 1) + 6, j = 5, i ≥ 2

22. A Labeled 5-Hairy Cycle
1
4
5
6
2
3
23
21
22
24
19
20
17
15
16
18
13
14
11
9
10
12
7
8

23. 7-Hairy Cycle
Theorem
For all n ≥ 3, Cn
S7
has a prime labeling.
Labeling
Let c1, c2, . . . cn
denote the vertices of Cn
, and let the pendant
be denoted pj
i
, 1 ≤ j ≤ 7. The labeling
formula f : V → {1, 2, . . . 8n} is given by:
f(c1) = 1
f(pj
1
) = j + 1
f(ci) =

8i − 5 i ≡ 2, 3, 6, 8, 9, 11, 12, 14 (mod 15)
8i − 3 i ≡ 4, 5, 7, 10, 13 (mod 15)
8i − 1 i ≡ 0, 1 (mod 15)
f(pj
i
) ∈ {8i − 7, 8i − 6, . . . , 8i} \ {f(ci)}

24. A Labeled 7-Hairy Cycle
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

25. Future Work
This technique works well for odd m values, so extending this
method to evens is an obvious goal.
However, this method will fail for m ≥ 16, due to a result by
S.S. Pillai, who found that in a string of 17 or more consecutive
integers, there will not always be an integer in the set relatively
prime to the rest.

26. Bertrand’s Postulate
Prior to this, labelings were constructed for graphs. However, for
this graph, a labeling scheme was devised ﬁrst.
Bertrand’s Postulate
For every n > 1, there exists a prime p such that n < p < 2n.
Unlike hairy cycles, this constructed graph will not have a
consistent number of pendants per cycle vertex. Instead, the
number will grow to allow us to use Bertrand’s postulate to ﬁnd
primes.

27. Bertrand Weed Graph
Earlier, the natural numbers were split up into sets of a
consistent size. For this graph, each set of integers needs to be
twice as large as the previous set.
Due to Bertrand’s postulate, there is guaranteed to be a prime
in each set. Assign the largest prime in each set as the label to
the cycle vertex, then label the pendant vertices with the rest
of the consecutive integers.

28. A Labeled Bertrand Weed Graph
1
2
13
10
9
11
14
7
12
8
5
4
3
6

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