Birank Number of Operationally Constructed Graphs

Andrew Benson

Advisor: Dr. Michael Fraboni

Liaison: Dr. Edward Roeder

Moravian College

Bethlehem, Pennsylvania

2013

1

ABSTRACT

A birank on a graph

g(a) = g(b), that

G

is a function

then on any path from

g(s) < g(a) < g(`).

g : V (G) → {1, 2, 3, ..., n}

a→b

there exist vertices

such that if

zl

and

`

such

This study was focused on examining the birank

number of graphs that could be created operationally through the process of the Cartesian product and single edge addition which is dened as the addition of an edge between two vertices of two arbitrary graphs (specic to this thesis). We constructed both upper and lower bounds on the birank number. In some cases, these bounds converged and allow for a denite birank number to be concluded. When bounds do not converge, the lowest upper and highest lower bounds are noted.

2

Contents

1 Introduction

7

1.1

Vertex Coloring . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

k -ranking

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.3

Denition of a Birank . . . . . . . . . . . . . . . . . . . . . . .

13

1.4

Looking Ahead

16

. . . . . . . . . . . . . . . . . . . . . . . . . .

2 Previous Work 2.1

2.2

Helpful Results

8

18 . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.1.1

Upper Bounds . . . . . . . . . . . . . . . . . . . . . . .

18

2.1.2

Lower Bound

19

2.1.3

Distance between Vertices

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Previous Work with Common Types of Graphs

20

. . . . . . . .

21

2.2.1

Path Graphs . . . . . . . . . . . . . . . . . . . . . . . .

21

2.2.2

Cycle Graphs

. . . . . . . . . . . . . . . . . . . . . . .

22

2.2.3

Complete Graphs . . . . . . . . . . . . . . . . . . . . .

23

2.2.4

Ladder Graphs

24

. . . . . . . . . . . . . . . . . . . . . .

3 Single Edge Addition of Vertex Transitive Graphs 3

26

3.1

General Graph Addition

. . . . . . . . . . . . . . . . . . . . .

28

3.2

General Graphs with Equal Birank Numbers . . . . . . . . . .

32

3.3

Single Edge Addition with Complete Graphs . . . . . . . . . .

33

4 Cartesian Products of Graphs

35

4.1

Notable Graphs . . . . . . . . . . . . . . . . . . . . . . . . . .

37

4.2

General Bounds . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4.3

Tighter Bounds . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.4

Rook's Graphs

44

4.5

Future work on

. . . . . . . . . . . . . . . . . . . . . . . . . .

Pn Pm

. . . . . . . . . . . . . . . . . . . . . .

5 Conclusion

45

47

4

List of Figures

1.1

Graph with 5-coloring and 4-coloring. . . . . . . . . . . . . . .

10

1.2

Same graph as Figure 1.1 with 3-coloring.

. . . . . . . . . . .

11

1.3

Graph

k -ranked

. . . . . . . . . . . . .

13

1.4

Graph

G.

k=7

with

and

k = 5.

Graph on left has the vertex set labeled. Graph on

right is labeled with the function

g(ua ) = a.

. . . . . . . . . .

14

2.1

P4

with an optimial birank.

. . . . . . . . . . . . . . . . . . .

22

2.2

C6

with an optimial birank.

. . . . . . . . . . . . . . . . . . .

23

2.3

K5

with an optimial birank.

. . . . . . . . . . . . . . . . . . .

24

2.4

L5

with an optimial birank.

. . . . . . . . . . . . . . . . . . .

25

3.1

Single edge addition of

P3

and

P1 .

3.2

Single edge addition of

C4

and

C3 . .

3.3

C 4 ⊕ P1

4.1

P 2 , P3 ,

and their Cartesian product

P2 P3 .

. . . . . . . . . .

36

4.2

P 3 , P4 ,

and their Cartesian product

P3 P4 .

. . . . . . . . . .

36

4.3

K 3 , K4 ,

. . . . . . . . . . . . . .

38

4.4

m

and

C 4 ⊕ P2 .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

. . . . . . . . . . . . . . . . . . . . . . .

31

and their Cartesian product.

copies of

G

and

n

27

copies of

5

H

making

Hm Gn .

. . . . . . .

39

4.5

P3 P3

biranked with 7 numbers.

6

. . . . . . . . . . . . . . . .

43

1

||

Introduction

Optimizing is one of the most important practices in modern day science and technology. It shows up in various subject texts, whether it be the best use of a data structure in computer science or how to get the most of a specic drug into the blood stream.

Although many algorithms have been developed to

settle the dispute of select tasks, there still exists many unsolved optimization problems.

As solutions of these problems are worked towards, newer topics

are introduced and thus more unsolved problems arise. In this study, a new graph labeling system is looked at for optimization. Many real world problems such as the best placement of communication towers or how ights connect a network of airports can be reduced to graphs. Graphs are everywhere, even in places one would not expect. These graphs can be of the simple types, such as the arrangement of desks in a classroom, to more complex networks, such as all the circuitry in a Boeing 747. Eventually it became necessary to study graphs and graph theory was introduced. Graph theory is a eld of mathematics that studies the structure of graphs.

In a

complex network, such as that listed previously, optimization is needed for many dierent reasons (cost and space are two of many driving factors). This

7

problem is how the optimization of ranking graphs, a particularly interesting subeld of graph theory, comes into play. Ranking graphs begins with placing dierent constraints on the labeling of a graph. The upcoming section, as well as the rest of this thesis, will require a simple, yet distinct, denition to answer the question of, What is a graph?

Denition 1. set

E(G),

A graph

G

is a triple consisting of a vertex set

V (G),

an edge

and a relation that associates with each edge two vertices called its

endpoints. [9]

Simply put, a graph is a collection of vertices with edges connecting them. Graphs can come in all shapes and sizes. A single graph can have many representations of it, much like the number

4

can be represented as

22

or

√ 16.

Though these representations may look completely dierent, as long as the same vertex set, edge set, and relation that associates them is the same, they are equivalent graphs. Now that a graph is dened, dierent forms of optimization on graphs will be looked at as background information. Previous work with graph labeling will be introduced before dening a birank and reviewing related literature.

1.1 Vertex Coloring Vertex coloring is an old idea dating back to the 1850s. At this time, Francis Guthrie was coloring a map of England and noticed that only four colors were needed so that two adjacent territories were dierent colors [8].

Excited by

this nd, he hypothesized that any map could be colored with only four colors.

8

It was not until over a century later that this hypothesis was proven to be true. Graph coloring is a well researched form of placing colors on the vertices of a graph. Traditionally these vertices are still represented as colors, but can also be represented by values placed on the vertex by a function. Using values allows for a more formal denition.

Denition 2. A k-coloring on a graph G is a labeling g : V (G) → {1, 2, 3, ..., k} such that if vertices

u, v

are adjacent,

g(u) 6= g(v).

To look at the map of England (or any map for that matter) as if it were a graph, each vertex would represent one territory while each shared edge would represent two territories being adjacent to one another. Once the map/graph is made, one can color the vertices as described above. As mathematicians began to explore Guthrie's problem, the rst task was to prove his original hypothesis, that any graph can be colored with 4 dierent colors. This optimization problem had multiple theorems and proofs written about it, many of which had fallacies that were revealed in years following their publication. The ve color theorem was one valid proof that contributed to this area of study. Figure 1.1 shows a graph with a amount of colors needed) such that

k -coloring (k

being the

k = 5.

This is a valid coloring on that particular graph and for almost a century, the ve color theorem was the smallest upper bound on the minimal

k -coloring.

This theorem was proven after failed attempts to prove the proposed four color theorem. Ironically, it is now practically obsolete because the four color theorem, a smaller upper bound, has been successfully proven.

9

Figure 1.1: Graph with 5-coloring and 4-coloring.

Theorem 1.

Every graph that can be drawn on a plane without intersecting

edges has a 4-coloring. [1]

The gure on the right of Figure 1.1 shows the same graph, but with a four coloring function. As long as one representation of the graph can fulll the requirement of not having any intersecting edges when drawn on a plane, then the graph has a 4-coloring. This theorem was a breakthrough because it solved a problem that had been worked on for over a century, nally conrming that every plane graph can be colored such that only four colors are required. This theorem, though interesting, provides only an upper bound for all graphs.

Therefore, it is not optimal in come cases.

In Figure 1.2, a graph

is created that has a 3-coloring on it. This graph could have a 4-coloring as Theorem 1 states and Figure 1.1 shows, but 3 colors will suce. The color of a vertex is dependent on the colors of the vertices that are

10

Figure 1.2: Same graph as Figure 1.1 with 3-coloring.

directly adjacent to that vertex. It makes no connection between the localized colors and far colors. To a degree a vertex that is very far could aect what color another vertex is, albeit not directly. Any traversal from one vertex to another through a vertex

→

edge

→

vertex combination is deemed a path.

By use of path, it is clear that vertices are slightly more dependent upon one another.

Denition 3. A path p from vertex a to vertex b is a list of vertices {u1 , u2 , u3 , ..., un } such that

ui

is adjacent to

ui+1 , a = u1

and

b = un .

Paths from one vertex to another are the basis for the next ranking system that will be described.

11

1.2 k-ranking Introduced in 1988, ranking is another form of labeling the vertices of a graph with a focus on nding the optimal way to rank the graph under self regulated constraints [5]. Ranked vertices depend upon all other vertices that have the same label throughout the graph, rather then strictly adjacent vertices.

k -ranking

places a label

1→k

A

on each vertex such that if two vertices have

the same rank, then any path between them contains a larger rank.

Denition 4. A k-ranking on a graph G is a labeling g : V (G) → {1, 2, 3, ..., k} such that if

g(u) = g(v),

any path

p

from

u→v

contains a vertex

`

such that

g(u) < g(`). As an example, given Figure 1.3, one can see it is possible to label the vertices of a graph such that these parameters are met for a the graph.

Both

k -ranking, k = 7

being more optimal.

and

k = 5,

To achieve an optimal

are valid with

k -ranking,

k -ranking k = 5

on

clearly

however, one would

need to nd the minimum number of ranks needed to create a

k -ranking

on a

particular graph. Ranks can be replaced such that they more closely follow to rules set forth by Denition 4 (i.e. the then the

k -ranking

k -ranking

on the right uses less values

on the left).

Though not as much research has gone into

k -ranking

graphs, some inter-

esting results have been reported. For one, given any path graph, its

k -ranking

number can be computed. There exists a function that yields an optimal value for

k,

such that

properly

k -rank

k

is equal to the least amount of ranks that could be used to

the path graph.

12

Figure 1.3: Graph

k -ranked

with

k=7

and

k = 5.

Theorem 2. A path Pn has a k-ranking number such that k = blog2 nc +1 [2]. Since

k -ranking, other forms of ranking have been proposed, such as corank-

ing and even more recently, biranking. Biranking, since its inception 10 years ago, has very little research and as such is the main topic for this thesis.

1.3 Denition of a Birank A biranking function, much like a

k -ranking function, requires a larger rank ex-

ist between two vertices ranked the same. The dierence is that a birank adds an additional constraint requiring a smaller value to also be placed between the vertices with the same rank.

Denition 5. whenever

Given a graph

g(a) = g(b),

g(a) > g(s)

and

G,

a function

on all paths

a→b

g : V (G) → N

is a birank on

there exist vertices

s

and

`

G

if

such that

g(a) < g(`).

Note that a graph always has a lot of birank functions. In order to create a birank function, when given a graph, label the vertices

13

{u1 , u2 , u3 , ..., un }.

From here, dene a function

g : V (G) → {1, 2, 3, ..., n}

by

g(ua ) = a.

Please

see Figure 1.4 for a simple depiction of this mapping function.

Figure 1.4: Graph

G.

Graph on left has the vertex set labeled. Graph on right

is labeled with the function

g(ua ) = a.

Notice that this mapping is a valid birank that uses the values from through

n where, again, n is the number of vertices in G.

1

This specic function

is called a trivial birank function. Biranking is one of the newest and a very useful ways to rank graphs. One practical application is in very large scale integration (VLSI). In VLSI, millions of electrical components are placed onto one small chip. The challenge is to arrange components to take the least amount of space but still function properly in the circuit. The architecture of the circuit is very complex, facing many stipulations to t each part functionally in a small space. example would be if components such that and

C

A

A, B,

and

C

A simple

all had dierent characteristics

dissipates an exorbitant amount of heat,

B

absorbs local heat,

was a component that assists in the circulation of air. What would be

the most sensible way to place these components if a circuit needed 2 of type

A,

and 1 of each

B

and

C?

It would not make sense to place

A

and

A

close

together since the heat in their local area would be way large. It is possible

14

to use

B

to absorb heat and

this issue.

Therefore

C

to help the ow of air from

A → B → C → A

A

to

B

to mediate

would be the most logical way to

place the four components. Not surprisingly, this pattern is identical to an optimal birank function on a four vertex path graph, labeled

2 → 1 → 3 → 2.

There are many functions that could properly birank a graph, and thus there are many valid biranks on any graph graphs

G

G.

The largest birank of any

would be a function that labels each vertex with a unique rank, like

that noted in Figure 1.4.

This function,

g(ua ) = a,

will birank any graph

because each vertex will have a unique rank, making it an upper bound on the birank number. This idea will be returned to later. Though there are many dierent functions that would allow for a valid birank on a graph, only a select few yield an optimal birank.

Denition 6.

A function

optimal if for all biranks

g : V (G) → {1, 2, ..., n}

f : V (G) → {1, 2, ..., r}

on

on

G

that is a birank is

G, n ≤ r.

To nd an optimal birank, a proof will be necessary. Once this birank is found, one can use following notation to identify it.

Denition 7.

Given a graph

G,

the birank number

number of ranks used in an optimal birank of

bi(G) = n

where

n

is the

G.

This allows for a simple, function independent value for the birank number. As stated above, there may be many dierent functions that will yield an optimal birank, but only one value for the birank number.

Since only one

of these many functions is needed to obtain a birank number, this thesis will

15

focus on nding

bi(G)

optimal birank on

rather then searching for all the functions that are an

G.

Throughout this thesis, there will be instances where the birank number of a graph can be found and proven. There will also be many instances when a birank number seems impossible to obtain. To work through this problem, an upper or lower bound will be dened. These bounds are very useful, not only as proofs, but eventually as the bounds improve, they will converge to one number, allowing for a birank number to be proven for a graph.

1.4 Looking Ahead Ladder graphs are one such graph that has been assessed for a birank number. Noting that ladder graphs are constructed by the Cartesian product of two dierent subgraphs, that is

P2 Pn

where

n

is the length of the ladder, it is

clear research on general graphs created by the Cartesian product should be conducted. In addition to this construct, another form of graph construction will also be considered. This construction, noted as Single Edge Addition, is specic to this paper and included because of the headway it makes in methods of proving the birank number of specic constructed graphs. Chapter 2 with discuss previous work done in the eld of biranking. Not much previous work has been conducted, but the work that was done is very signicant. Chapter 3 will introduce the idea of stitching two graphs together by means of adding a single edge between any of its two vertices.

Finally,

Chapter 4 will discuss a common topic of graph theory, the Cartesian product

16

of two graphs, and will place bounds on the birank number of cross product graphs.

17

2

||

Previous Work

2.1 Helpful Results Now that the basic information is covered, there are a couple results regarding biranks that may seem obvious to the reader. It is worth pointing these out in detail before continuing on to a discussion of previous work done with dierent types of graphs. The following section will discuss denitions and theories that will be used later on in the paper. The proofs and how they relate to biranking are very important.

2.1.1

Upper Bounds

There are dierent ways to obtain the birank number of a specic graph or graph family. One of which is to construct both upper and lower bounds, then work towards convergence. Both of these bounds can be proven in dierent ways, some being more dicult and requiring more clever proofs than others. To obtain an upper bound for the birank number of a graph, a birank function must be looked for on the graph. For small graphs, graphs that can be represented on paper with vertices and edges, this function could be found

18

by trial and error, making sure that any path

g(`).

a→b

contains both

g(s)

and

For larger or general graphs, this method of manually testing dierent

functions does not work as eectively. Instead, a general function will need to be proposed and later proven to be a birank. Regardless of method, trial and error or theorizing and proving, the amount of labels necessary will be an upper bound on the birank number.

2.1.2

Lower Bound

Unlike upper bounds where a function is an upper bound on the birank number if it can be proven to be a birank, a lower bound on the birank number must show that is is impossible for a birank to exist with a set number of labels. The most notable and simplest way to nd a lower bound is to use a subgraph.

Denition 8.

A graph

H

is a subgraph of graph

G

if

V (H) ⊆ V (G)

and

E(H) ⊆ E(G). The following useful theorem, which appears in Fisher, et al. [3], relates a lower bound of the birank number of a graph to the birank number of its subgraphs.

Theorem 3. Given graphs G, H , if H is a subgraph of G, then bi(H) ≤ bi(G). [3]

Proof. Graph

G

can be constructed from

H

by adding edges and vertices.

No matter how vertices and edges are added, there will be no change in the original paths between vertices in subgraph requires

bi(H)

ranks, making

H,

bi(G) ≥ bi(H). 19

and therefore subgraph

H

still

2.1.3

Distance between Vertices

As stated in Denition 5, for a birank function on a graph to assign two vertices the same value, there must exist both a smaller and larger rank on any path between the two vertices. This gives a lower limit on the distance between vertices with equal ranks. The lemma below will show that at least two vertices are needed between any vertices with the same rank.

Lemma 1. g(b),

G,

Given a graph

an edge from

a → b,

g(a) =

let

g

be a birank on

then

g

is not a birank because it is impossible for larger

must include at least two other vertices.

and smaller ranks to exist between that there are edges

a birank, if

to

G,

b

g : V (G) → N,

a

then every path from

Proof. Given graph

a function

a → m → b,

a

then

and

g

G

b.

and

a

and

b

If there exists

If there exists a vertex

m

such

is not a birank because vertices with

larger and smaller ranks can not exist between be at least two vertices between

g(a) = g(b).

for

g

a and b.

Therefore, there must

to be a birank.

Next, a word is needed to dene the space between these vertices. Rather then stating at least two vertices are necessary on all paths between another two vertices with repeated ranks, the word distance will be used.

Denition 9.

Distance between two vertices is the number of edges along the

shortest path between the vertices.

Distance is a parameter that measures not only the number of edges between two vertices, but the number of vertices as well. By Lemma 1, one can

20

see that for birank

g

on

G

and

g(a) = g(b),

tance three (three edges, two vertices) from

b

vertex for

g

a

must be at least dis-

to be a birank on

G.

This

argument is useful in proofs with graphs that do not always meet the initial requirement of distance.

2.2 Previous Work with Common Types of Graphs As previously stated, biranking of graphs is only now growing is popularity, allowing for a multitude of directions for research. Though there are innitely many types of graphs ranging from singular points on a plane to a complex network of vertices and edges forming a three dimensional gure, previous research was conducted on basic families which could easily be drawn and modeled by hand.

Due to the simplicity, trial and error could be used to

obtain a general idea of what type of function would fair well with regards to an optimal birank. Eventually this modeling would lead to ideas that would spark intuition to create a proof of the optimal birank of a specic graph family.

2.2.1

Path Graphs

A path is a family of simple graphs whose vertices can be ordered such that two vertices are adjacent if they are consecutive in the list [9]. This means that all vertices are of degree two except for

v1

the path. Paths are usually denoted as

and

Pn

vn , the rst and last vertices on

where

n

is the number of vertices

used in the path. Figure 2.1 below shows us a path of 4 vertices and labels

21

creating a birank on

P4 .

Figure 2.1:

P4

with an optimial birank.

The next question would be if this was an optimal birank. Previous work by Fisher et al. [3] supports the birank number shown above and reveals that the birank number of any path graphs with

n

vertices can be computed.

Theorem 4. bi(Pn ) = blog2 (n + 1)c + blog2 (n + 1 − (2blog2 nc−1 ))c If

3.

n = 4 is evaluated in the equation of Theorem 4, it is clear that bi(P4 ) =

A function to birank

P4

with only three values is shown in Figure 2.1.

This gure is consistent with the example stated earlier in Chapter 1 (use of biranking in VLSI). It is also the smallest path length such that

bi(Pn ) 6= n.

Adding a single edge between the rst and last vertex in a path graph will create a new type of graph, noted as a cycle, which was also explored in previous research.

2.2.2

Cycle Graphs

A cycle graph is a graph with equal numbers of vertices and edges such that each vertex has two connecting edges and there exists a path from

vn

to itself

such that every vertex is passed through exactly once. These types of graph can be easily seen as paths that have an edge from where

n

v1

and

vn .

Cycles, denoted

Cn

is the number of vertices, become useful in this study when modeling

and discussing theorems in Chapter 3.

22

From previous work, again by Fisher et al. [3], the birank number of any cycle graph is computed by the following formula.

Theorem 5. bi(Cn ) = bi(Pn−2 ) + 2 It is very interesting that the concluding equation to disclose the birank of any cycle can be written such that it is directly dependent upon that of a path graph with two less vertices. As an example, Figure 2.2 shows

C6

with

an optimal birank.

Figure 2.2:

C6

with an optimial birank.

Evaluating the equation in Theorem 5 with 4 of course) shows that

bi(C6 ) = 5. C6 ,

n = 6 (making use of Theorem

much like

because it is the smallest in its family such that

2.2.3

P4 ,

bi(Cn ) 6= n.

Complete Graphs

Complete graphs are graphs such that for all vertices an edge between vertex is with

is a notable graph

n − 1,

va

and

vb .

va

Given a number of vertices

meaning it has

n−1

n vertices are denoted as Kn .

and

n,

vb

there exists

the degree of each

edges connecting to it. Complete graphs

Figure 2.3 shows an example of a complete

graph with 5 vertices.

23

Figure 2.3:

K5

with an optimial birank.

The birank number of a complete graph is

bi(Kn ) = n.

This is because

all of the vertices are distance 1 from each other, meaning all vertices have an edge connecting them. If there were a repeated rank, there would be a direct path between the two ranks invalidating it as a birank.

2.2.4

Ladder Graphs

More recently work has been done to uncover the birank of any ladder graph. A ladder graph can be modeled as two path graphs of the same length with connecting edges between

ui

and

ble a ladder and is denoted as

vj

if and only if

Ln .

i = j.

This graph will resem-

Previous research conducted by Fraboni

et al. [4] yields a result for the optimal birank of a ladder of any length. The result is shown below.

c + blog2 n+1 c + blog2 n+1 c + blog2 n+1 c+5 Theorem 6. bi(Ln ) = blog2 n+1 2 3 5 7 As an example, Figure 2.4 shows a ladder graph of length 5 with a birank function. Evaluating Theorem 6 for

n=5

with yield

bi(L5 ) = 7,

making the

function on Figure 2.4 an optimal birank. Ladder graphs, unlike paths, cycles, and complete graphs, are a very basic operationally constructed graph. They are made by the common operation of

24

Figure 2.4:

L5

with an optimial birank.

the Cartesian product of graphs, specically

P2 Pm .

Though at the time when

Theorem 6 was proven, the authors make no note of focusing on ladder graphs as operationally constructed, rather, it seems like ladder graphs are just the next logical step. Thinking of this graph as operationally constructed opens many doors for continued research. Section 4.5 looks at the future research that stems from work with ladder graphs.

25

3

||

Single Edge Addition of Vertex Transitive Graphs

The rst form of graph construction considered in the thesis will be one that was created just for this use. Single edge addition is a unique process of adding graphs together. This process creates a new graph by adding an edge and using a vertex in each graph as endpoints. Though this method is dened specically for this thesis, it may someday be useful in the greater eld of graph theory. As one can probably recognize, especially with graphs that have many vertices, there is a large but nite number of ways to connect two graphs. For graphs

Gn

and

Hm

where

to add an edge from

Gn

n, m are the number of vertices, there are n · m ways

to

Hm .

Depending on the properties of

Gn

and

Hm ,

the product graphs could have very interesting representations and some will be duplicates of other graphs produced by the same

Gn

3.1, one can see that by adding a single edge between

P3

and and

Hm .

In Figure

P1 ,

more than

one new graph is possible. The gure shows the two distinct possible graphs, with the rst graph (P4 ) having a multiplicity of 2. Of the three possibilities, only two are unique. Interestingly enough, the

26

Figure 3.1: Single edge addition of

P3

and

P1 .

birank number of the two unique graphs dier. The rst unique graph which creates

P4

has a birank number of 3 by Theorem 4 and the three pointed star

graph has a trivial birank of 4 (by distance argument). As seen above, single edge addition is not well dened for arbitrary graphs. Therefore, no denite conjecture can be made about the single edge addition of general graphs until this addition is dened better. For this addition to be well dened, single edge addition must be limited to vertex transitive graphs.

Denition 10.

A graph

an automorphism

Requiring dened. Take

f

of

G and H C4

and

G

G

is vertex transitive if for every

such that

v1 , v2 ∈ G

there is

f (v1 ) = v2 .

to be vertex transitive makes single edge addition well

C3

for example. Both are vertex transitive graphs, and

therefore should be well dened under single edge addition.

There are still

12 dierent ways to connect the two graphs by adding a single edge, but all product graphs are isomorphic. This constructed graph can be seen in Figure 3.2. If a cyclic function

f

is placed on

C4 such that f (A) = B, f (B) = C, f (C) =

D, and f (D) = A, the same product graph would exist even though the vertices 27

Figure 3.2: Single edge addition of

C4

and

C3 .

are labeled dierently. Single edge addition of vertex transitive graphs is now well dened and a symbol is needed to denote this operation. Please note this symbol is specic to this text. Given

G, H

vertex transitive graphs,

the graph constructed by adding a single edge between a vertex in vertex in

G⊕H G

is

and a

H.

Denition 11. G ⊕ H

is the graph with vertex set

E(G) ∪ E(H) ∪ {(a, b)}

for some

a∈G

and

V (G) ∪ V (H)

and edge set

b ∈ H.

3.1 General Graph Addition The rst proof in this chapter describes an upper bound of the single edge addition of two general vertex transitive graphs. To show this upper bound, a

G

is still a birank

f +k

is also a birank

lemma will be needed. This lemma will prove a birank on if every rank is increased by a constant value.

Lemma 2. on

If

f

is a birank on graph

G

and

k ∈ N,

then

G.

Proof. Given

Given

a, b

G

with

and

f,

a birank on

f 0 (a) = f 0 (b),

then

G,

let

f 0 : V (G) → N

f (a) = f (b). 28

by

f 0 (c) = f (c) + k .

Given any path from

a

to

b

in G, there exist vertices

f 0 (cs ) < f 0 (ca ) < f 0 (c` ).

cs

c`

and

Therefore

f (cs ) < f (a) < f (c` )

such that

f (c) + k

is also birank on graph

and so

G.

Lemma 2 will now be used in a proof of an upper bound for a general graph

G ⊕ H.

Theorem 7.

For vertex transitive graphs

G, H

with

bi(G) ≥ bi(H), bi(G ⊕

H) ≤ bi(G) + 1. Proof. Label the vertices of graphs

and

V (H) = {v1 , v2 , ..., vm }.

an optimal birank on in

G

(say

Dene

us

with

H.

v`

g(us ) = 1)

f : V (G ⊕ H) → N

be an optimal birank birank on

G and h be

to the largest in

such that if

This makes

H

(say

v`

with

t ∈ G, f (t) = g(t)

f (us ) = 1

h(v` ) = bi(H)).

and if

t ∈ H, f (t) =

and

f (v` ) = bi(G) + 1

f (ta ) = f (tb ),

three cases exist:

with

us

at either end of the new edge.

Given vertices Case 1: Both

f = g path

g

V (G) = {u1 , u2 , ..., un }

such that

Create an edge from the vertex with the smallest rank

h(t) + bi(G) − bi(H) + 1. and

Let

G and H

p0

on

G

ta , tb ∈ G ⊕ H

ta f

so

and

tb

are in

with

G.

is a birank on

completely on

G

G.

Any path

p

from

ta → tb

contains a

(if a path contains the new edge, it must reenter

through the same vertex).

Therefore, there exists

ts

and

t`

on

p0

such that

f (ts ) < f (ta ) < f (t` ). Case 2: Both

f = h+k path

on

H

ta

and

where

p from ta → tb

k

tb

are in

H.

is a constant. By Lemma 2,

contains a path

p0

completely on

f

is a birank on

H.

H

(if a path contains the

Any

new edge, it must reenter through the same vertex). Therefore, there exists

29

ts

and

t`

on

p0

such that

ta

Case 3:

ta ∈ G

and

and

tb

f (ts ) < f (ta ) < f (t` ). are in dierent components, without loss of generality,

tb ∈ H .

Every path between

and

tb ∈ H

must contain the added edge which

ts ∈ G, f (ts ) = g(vs ) = 1

links

ts

Since

t` ∈ H , f (t` ) = h(v` ) + bi(G) − bi(H) + 1 = bi(G) + 1,

to

t` .

ta ∈ G

bi(G) ≥ f (ta ).

Since

Therefore any path

p

from

and therefore,

ta → tb

with

f (ts ) < f (ta ).

therefore

ta ∈ G

and

f (t` ) >

tb ∈ H

has

vertices with both larger and smaller ranks. Since all paths between vertices ta , tb for with larger and smaller ranks, the function value of

f

is

f (t` ) = bi(G) + 1

f (ta ) = f (tb ) contain both vertices

f

and therefore

is a birank on

G⊕H .

The largest

bi(G ⊕ H) ≤ bi(G) + 1.

Theorem 7 shows the tightest upper bound that was found to exist for general vertex transitive graphs. This bound can be thought of as relatively good, as it only allows for a small region above the actual birank number of one of the two graphs. Constructing a lower bound would be the next step to nding the birank number of

G ⊕ H.

A subgraph argument will be used to

show that the upper and lower bound only dier by 1.

Theorem 8.

For vertex transitive graphs

G, H

with

bi(G) ≥ bi(H), bi(G ⊕

H) ≥ bi(G). Proof. The graph

G⊕H

has both

G and H

as subgraphs. A graph can not have

a smaller birank number then any of its subgraphs. Therefore,

bi(G ⊕ H) ≥

bi(G). A subgraph argument may seem elementary, but it can be a vital tool when

30

it comes to a proof of a lower bound for the birank number. A summary of these two theorems is below.

Corollary 1.

For general vertex transitive graphs

bi(G ⊕ H) = bi(G)

or

G, H

with

bi(G) ≥ bi(H),

bi(G ⊕ H) = bi(G) + 1.

As an example of how easily the birank number can change, consider two constructed graphs,

C 4 ⊕ P1

and

C 4 ⊕ P2 .

Figure 3.3 shows these graphs with

functions for an optimal birank.

Figure 3.3:

The graph on the left, of Theorem 8 and so

C4 ⊕ P 1

C 4 ⊕ P1 ,

and

C4 ⊕ P 2 .

yields a birank number at the lower bound

bi(C4 ⊕ P1 ) = bi(C4 ) = 4.

By exchanging

P2

for

P1 ,

the

birank number increases to that noted by the upper bound of Theorem 7, such that

bi(C4 ⊕ P1 ) ≤ bi(C4 ) + 1 = 5.

how

C4

A diligent mind can see that no matter

is translated, only the rank 2 can be repeated on the connected graph

to be a valid birank, making

bi(C4 ⊕ P2 ) ≥ bi(C4 ) + 1 = 5.

Corollary 1 shows the tightest bounds constructed for single edge addition of vertex transitive graphs. A more general theorem than those listed was not found, but optimal biranks for specic graphs were found. Looking at some specic cases now will lead to more generalizations in future research.

31

3.2 General Graphs with Equal Birank Numbers Consider

bi(G ⊕ H)

two subgraphs

1

through

that ranks

n.

G, H

for the special case that

bi(G) = bi(H) = n.

to have optimal biranks of

n,

both graphs require ranks

When adding a single edge between these two graphs, the fact

1 through n are used on both poses a problem.

explains why

For the

The following proof

bi(G ⊕ H) 6= bi(G).

Theorem 9.

For vertex transitive graphs

G, H

with

bi(G) = bi(H), bi(G ⊕

H) > bi(G). Proof. Consider vertex transitive graphs

Both

G

and

H

are subgraphs of

G, H

G⊕H

such that

bi(G) = bi(H) = n.

and will require

n

distinct labels.

The largest and smallest values on the graph, can not appear more than once meaning more then

n

labels are needed. Therefore,

bi(G ⊕ H) > bi(G).

Combining Theorem 9 with Theorem 7, it is clear for

bi(G ⊕ H) = bi(G) + 1

bi(G) = bi(H).

Corollary 2.

For vertex transitive graphs

G, H

with

bi(G) = bi(H), bi(G ⊕

H) = bi(G) + 1 Theorem 9 along with the Corollary allows for a way to compute the birank number of two like systems that was combined through single edge addition. This also leads to an easy solution for a seemingly dicult problem. Consider a situation where there exists graph

G with bi(G) known. 32

One is to nd graph

H

such that

bi(G ⊕ H) = bi(G) + 1.

thought, would be to make

H = Kn

An obvious solution, along with minimal such that

bi(G) = n

because

bi(Kn ) = n.

Although there are many solutions that would solve this problem, this is by far the most simple. This example is just one of many that involves complete graphs.

In the

next section, a theorem that makes use of the properties of complete graphs will be considered.

3.3 Single Edge Addition with Complete Graphs Complete graphs are a special family of graphs that are also vertex transitive. The geometric properties of a complete graph allows for a theorem that provides the birank number in some cases. This theorem has to do with the distance of the vertices of

Theorem 10.

Kn .

Given a vertex transitive graph

G,

if

bi(Kn ) ≥ bi(G),

then

bi(Kn ⊕ G) = bi(Kn ) + 1. Proof. Consider

K n ⊕ K1 ,

note

Kn

required

n

labels. The placement of

through single edge addition makes it distance 2 or less from all vertices in This means that

K1 Kn .

K1 needs a unique label so bi(Kn ⊕K1 ) = bi(Kn )+1. Kn ⊕K1

is a subgraph of all graphs of the form

bi(Kn ⊕ G) = bi(Kn ) + 1

Kn ⊕ G.

Therefore, if

bi(Kn ) ≥ bi(G),

by Corollary 1.

Though the birank number obtained in the scenario mentioned is specic to complete graphs that have a greater birank number than the other graphs

33

used in single edge addition, it still demonstrates the power of where edges are placed on a graph. No general bounds have been found for cases of

bi(Kn ).

One can consider a case for both

bi(G) + 1.

Kn ⊕ G

such that

bi(Kn ⊕G) = bi(G) and bi(Kn ⊕G) =

Looking back to Figure 3.3, one can see that for

bi(K1 ⊕ C4 ) = bi(C4 )

as well as

For the specic case of

bi(G) >

bi(C4 ) > bi(Kn ),

bi(K2 ⊕ C4 ) = bi(C4 ) + 1.

Kn ⊕Km graphs, the general graph noted in Theorem

10 could be a complete graph

Km

such that

bi(Kn ) ≥ bi(Km ).

This leads to a

Corollary that is essentially a restatement of the theorem, with interchangeable

Kn

and

Km

depending on which has a greater birank.

Corollary 3.

For graphs

Kn , Km

with

n ≥ m, bi(Kn ⊕ Km ) = n + 1.

34

4

||

Cartesian Products of Graphs

While single edge addition is a very selective process that only pertains to vertex transitive graphs, the Cartesian product of two graphs is not limited by constraints of the factor graphs.

This allows for a multitude of product

graphs, each unique in its own way. To create the Cartesian product graph,

GH ,

start with general graphs

Denition 12. edge between between

v

The graph

(u, v)

and

v0

H

and

in

GH

(u0 , v 0 )

or

G

H.

and

if and only if

v = v0

V (G) × V (H)

has the vertex set

u = u0

and has an

and there exists and edge

and there exists an edge between

u

and

u0

in

G. This will essentially place a copy of graph

n

copies of

H,

H 1 , H 2 , ..., H n in

G, H

noted as

by

m

H 1 , H 2 , ..., H n )

copies of graph

G

H

at every vertex of

G

(making

and connect the vertices of graph

where

n, m

are the number of vertices

respectively.

A few basic examples of graphs created by the Cartesian product can be shown with path graphs. It can be seen that any path create a grid two high and

n

Pn

crossed with

P2

will

long. These types of graphs belong to a specic

family and are called ladder graphs as noted in Chapter 2. The birank number

35

of a ladder graph is already know and presented as Theorem 6. A simple example of a ladder graph can be seen below by taking the Cartesian product of

P2

and

Figure 4.1:

P3

yielding

P2 , P 3 ,

P2 P3 .

and their Cartesian product

P2 P3 .

Ladder graphs are a simple basis of a larger family of graphs. Figure 4.2 shows

P3 P4 ,

another example of a simple Cartesian product. Although this

graph is not a ladder graph, they both belong to a common greater family.

Figure 4.2:

P3 , P 4 ,

and their Cartesian product

This common family is generated by grid structures, known as a grid graph.

36

Pn Pm

P3 P4 .

yielding simple rectangular

4.1 Notable Graphs Now that the method of graph construction by the Cartesian product has been established any two graphs can be combined in such a way. Although there are innitely many creations, there are still specic families of created graphs that are worth taking the time to note.

As stated above, the Cartesian product

of two path graphs will be a grid graph. This is the greater family to which both

P2 P3

P3 P4

and

belong, making paths and ladders subfamilies of grid

graphs. Other graphs, such as the Cartesian product of

Cn

and

Pm , Cn Pm ,

noted as prism graphs. One representation of these graphs is to have centric cycle graphs connected by

n

m

are

con-

radial spokes.

Complete graphs create another notable family. This theorem is very specic to complete graphs due to the distance of the vertices. Although and

Kn Cm

do not have family names,

Denition 13.

Kn Km

The Cartesian product of

Kn

Kn Pm

has been named.

and

Km

is a rooks graph.

This name was coined from the rook chess piece because the edges of the graph show all allowable moves of a rook on a chess board (K8 K8 would describe these moves on a standard chess board). Though it may seem complex, Figure 4.3 shows one of the easiest rook's graph to visualize.

K3 K4

is also

C3 K4 (K3 ≡ C3 )

representation of

Kn Km

the gure shows how complicated a visual

become with any values of

Another visual representation of sions

n

by

m

Even though

Kn Km

with an edge existing between

37

n, m.

could be a vertex grid of dimen-

vn,m

and

vx,y

if

n=x

or

m = y.

Figure 4.3:

K3 , K4 ,

and their Cartesian product.

By this method, all vertices in the same row and column will be connected. Though there are many dierent graphs created through the Cartesian product of two graphs, one can see how complex these graphs can get even if only basic graphs are used. The remainder of the chapter will focus on proofs related to both general and specic types of Cartesian product graphs.

4.2 General Bounds As seen above, there are many dierent Cartesian product families of graphs created by even the most basic families of graphs. Because of this, it is very dicult to generalize an upper limit that will hold true for all graphs. This rst proof will explore an upper bound on the birank number of the Cartesian product of graphs

G

and

H.

Before stating the theorem and beginning the proof, it is important to make clear the specic construction and labeling system of the graph. When constructing

GH ,

consider

38

H

to be a path graph oriented on a

vertical plane and sider

HG

labeled if

r=i

ut .

G

to be a cycle graph oriented on a horizontal plane. Con-

by Denition 12. Let the vertices in Note that on

and

ut

HG, (pr , ut )

is adjacent to

uj

in

G

or

H

(pi , uj )

and

t=j

and

be labeled

pr

and

G

be

are adjacent if and only

pr

is adjacent to

pi

in

H.

Please see Figure 4.4 for an example of this set up.

Figure 4.4:

m

copies of

G

and

n

copies of

H

making

Hm Gn .

Next, a term is needed to be dened that will be used in the proof soon to follow. That term is projection.

Denition 14. Theorem 11.

(p, u) ∈ GH

The projection of a vertex

Given a graph

G

and graph

H

with

n

onto

G

vertices,

is

p ∈ G.

bi(HG) ≤

n · bi(G). Proof. Given two graphs,

{u1 , u2 , ..., um } on

G.

with

and

and

and

H,

label the vertices such that

V (H) = {p1 , p2 , ..., pn }.

Consider a function

pr ∈ H

G

f : HG → N

ut ∈ G. 39

Let

by

g

V (G) =

be optimal birank function

f ((pr , ut )) = n · g(ut ) − (r − 1)

Given two vertices note

r = i

c0

of

g(ut ) = g(uj ). on

c0

such that

(pa , u` )

and

(pi , uj )

and

c

Let

c

onto

Since

g

be a path between

G.

The path

c0

It is clear that

n

(pr , ut )

1 ≤ r ≤ n.

and

and

(pi , uj ).

is now a path from

ut

to

for some

a

and

g(ut ) < g(u` )

Since

c0

contains

us

and

Also,

Consider the

uj

in

is a birank on G, there exists some vertices

g(us ) < g(ut ) < g(u` ).

(pb , us )

f ((pr , ut )) = f ((pi , uj )),

such that

since the ranks are equivalent mod

g(uj ) = g(ut ). projection

(pr , ut )

G

u`

u` , c

where

and

us

contains

b.

and both sides are integers, therefore it is

possible to subtract 1 from the right and allow the possibility of the left to equal the right. Also recall that

n

is strictly positive:

g(ut ) ≤ g(u` ) − 1

n · g(ut ) ≤ n · g(u` ) − n By the same argument above, 1 can be added to the right and remove the possibility for the left and right to be equal:

n · g(ut ) < n · g(u` ) − n + 1

n · g(ut ) < n · g(u` ) − (n − 1) Substituting the equations from above, note that

f ((pa , u` )) ≥ n · g(u` ) − (n − 1):

f ((pr , ut )) < f ((pa , u` ))

40

f ((pr , ut )) ≤ n · g(ut )

and

Therefore, Similarly,

f ((pr , ut )) < f ((pa , u` )).

f ((pr , ut )) > f ((pb , us )).

Thus it is shown

f

is a birank on

HG

and

bi(HG) ≤ n · bi(G).

To clear any confusion of how some parts of this proof work, here is a simpler explanation of the function rst part of

f ((pr , ut )), ng(ut ),

connected by the same

H,

f ((pr , ut )) = n · g(ut ) − (r − 1).

This

will place a integral multiple on every vertex

that is, every vertex in individual subgraphs

will be ranked with the same value. The second part,

−(r − 1),

Hr

will subtract

a unique value for every disk graph. This value is dependent on the level of the disk. This level is also know as the vertex in

H (pr )

where the disk is

connected. This will cause each disk to have a unique set of values, with the top disk having values of

g(ut ), the disk below that having values of n·g(ut )−1,

and the bottom disk utilizing

ng(ut ) − m + 1,

vertices to have the same value (f ((pr , ut )) the same disk level (r

making it impossible for two

= f ((pi , uj )))

unless they are on

= i).

This is the most general proof showing an upper bound for Cartesian product graphs. It should also be noted that a corollary follows to make the result more precise.

Corollary 4. bi(HG) ≤ min{n · bi(G), m · bi(H)} where n is the number of vertices in

H

and

m

is the number of vertices in

41

G.

4.3 Tighter Bounds One of the most seemingly obvious relations between graphs is the dierence between of

bi(G) · bi(H)

bi(GH)

and

bi(G) · bi(H).

being an upper bound of

easily be proven wrong using

P4

and

bi(GH)

P4

G, H , and GH ,

At rst, the possibility

seems clear, but this could

if it is recalled that

After many trials, a combination that made

bi(P4 P4 ) ≤ 9

Therefore, a proof was compiled to show that

bi(P4 ) = 3.

was not obvious.

bi(P4 P4 ) > 9.

Theorem 12. bi(P4 P4 ) > 9. Proof. Assume

bi(P4 P4 ) ≤ 9.

Since

P4 P4

is a grid, the ranks of

1 and 2 can

only be placed once because there are not enough smaller values for them to be placed more then once. Similarly with argument that the rank ranks left. Therefore each. For

3

5

and

9.

One can see by a distance

can only be placed 4 times, leaving 8 vertices and 4

3, 4, 6

and

7

must be placed a total of 8 times, or twice

to be placed twice, it will be required to be placed in one corner

to ensure all paths contain either value

8

1 or 2.

Similarly,

7 must be in a corner.

5 needs to be placed in at least 3 corners. P4 P4

The

only has 4 corners and

therefore at least one additional vertex will need a unique rank.

Therefore,

bi(P4 P4 ) > 9. This example has shown that generality

bi(G) · bi(H)

the option of

bi(P4 P4 ) > bi(P4 ) · bi(P4 )

in not an upper bound for

bi(G) · bi(H)

being related to

and therefore, in

bi(GH).

bi(GH)

This still leaves

by other means, such as

a lower bound. The next example shows why this is not true either.

42

Consider the Cartesian product of

P3

and

P3 ,

bi(P3 ) · bi(P3 ) = 9.

note

Through a little manipulation, a signicantly smaller birank can be found. Figure 4.5 illustrates that

bi(P3 P4 ) ≤ 7.

Figure 4.5:

P3 P3

biranked with 7 numbers.

In this case, the upper bound given by Theorem 11 happens to be 9, along with the relation in question (also 9). A birank of 7 was achieved through trial and error. This counterexample shows that in generality, a lower bound of

is not

bi(GH).

A lower bound of

P3 P3

describes the perimeter of

7.

bi(G) · bi(H)

can be found from the subgraph of

P3 P3

and

bi(C8 ) = 6.

Therefore,

P3 P3 . C8

6 ≤ bi(P3 P3 ) ≤

This makes the bounds signicantly smaller while approaching the birank

number. Much like Theorem 12, it can be shown that

bi(P3 P3 ) = 7.

Theorem 13. bi(P3 P3 ) = 7 Proof. Assume

bi(P3 P3 ) = 6.

Let

n(x) represent the number of times rank x

can be used in a function to birank graph

n(6) = 1

P3 P3 . n(1) = n(2) = 1 and n(5) =

because nothing is smaller or larger then these values.

along with

n(4) ≤ 2 due to distance.

8 times on 9 vertices, making

Therefore, ranks

bi(P3 P3 ) 6= 6. 43

n(3) ≤ 2

1 → 6 can only be used

Therefore,

bi(P3 P3 ) = 7.

Theorem 13 shows that the lower bound constructed by a subgraph not optimal, and therefore the

bi(P3 P3 ) = 7.

Although this is not the preferred

method to obtaining the birank number (trial and error then removing lower bound) it still works if there is a single graph of interest. These examples show that there is no obvious relation between and

bi(GH).

bi(G)·bi(H)

This intuitive relation is nonexistent at this time, but may have

a more complex relation as research progresses.

4.4 Rook's Graphs As stated above, rook's graphs that are constructed by the Cartesian product of complete graphs have a theorem to themselves due to the distance of their vertices. A theorem and proof were constructed to show exact birank of any sized

Km Kn

graph.

Theorem 14. bi(Km Kn ) = m · n Proof. Any two vertices

therefore

(ur , vt )

f (ur , vt ) 6= f (ui , vj ),

and

(ui , vj )

making

on

Kn Km

are distance two and

bi(Kn Km ) = n · m.

This theorem relies on a distance argument which is very useful for complete graphs. Interestingly enough, Theorem 14 shows a perfect relationship between

bi(Kn ) · bi(Km )

and

bi(Kn Km ).

Corollary 5. bi(Km Kn ) = bi(Km ) · bi(Kn )

44

4.5 Future work on PnPm Though the Cartesian product of two graphs was not limited to specic types of graphs, this generality made bounds harder to achieve than those found in Chapter 3.

Finding a general rule is dicult when there are many possible

graphs that could come from the Cartesian product of general graphs. Starting with the information in [3], one can see the complexity of the general formula for an optimal birank on a path graph. Reviewing work conducted in [4] shows an even more complex approach to generalizing a family of graphs (which happen to be a Cartesian product graph).

These two sources conclude the

rst two subfamilies of grid graphs. Grid graphs are the Cartesian product of

Gn,m .

Stated above, paths (Pn

= Gn,1 )

Pn

and

Pm

and Ladders (Ln

such that

= Gn,2 )

Pn Pm =

are the rst

subfamilies of these graphs. From these families, the birank number of the rst two graphs in the subfamily

Gn,3

are known, that is

Theorem 13 yields the birank number for the subfamily of

G3,m

G3,3 .

P3 = G3,1

and

Therefore the rst

L3 = G3,2 .

3

graphs of

are known such that:

bi(G3,1 ) = 3 bi(G3,2 ) = 5 bi(G3,3 ) = 7 Looking back at Figure 4.2 (P3 P4

= G3,4 ), trial and error yielded a tighter

upper bound on the birank number then that given by Theorem 11. This upper

45

bound was found such that

bi(G3,4 ) ≤ 8.

birank number, the birank number of

Using the subgraph

G3,4

G3,3 with a known

is now bounded to

7 ≤ bi(G3,4 ) ≤ 8.

Theorem 15. bi(G3,4 ) = 8 Proof. Assume

x

bi(G3,4 ) = 7.

Let

n(x)

represent the number of times rank

can be used in a function to birank graph

n(6) = n(7) = 1. n(3) ≤ 2

G3,4 . n(1) = n(2) = 1

to ensure any path from two vertices of rank

pass through a smaller rank (two smaller ranks are needed (1, 2) if in a corner). Similarly,

4.

Therefore, ranks

bi(G3,4 ) 6= 7.

n(5) ≤ 2. n(4) ≤ 3

1→7

Therefore,

and

3

3

is place

due to the distance placement of

can only be used 11 times on 12 vertices, making

bi(G3,4 ) = 8.

This is a good start for generalizing the

G3,m

family. Although the current

proof method used could not be continued, something along the lines of that outlined in either [4] or [3] would make better headway towards generalization. of both

G3,m

and

Gn,m .

46

5

||

Conclusion

As stated in the Chapter 1, biranking is one of the youngest forms of graph ranking. Not much research has gone into the specics, and this paper, being one of the rst composed on the matter, makes good headway as a transition from specic families of graphs to more complex graphs. Chapter 2 discusses what research was previously done. This allows for a grasp of the concept of biranking and also lets the reader/researcher compute the birank number for specic graph rather the just state that the birank number could be found this way.

For example, given two path graphs with

known lengths, an upper and lower bound on the birank number of their Cartesian product can be known as an actual number rather then an arbitrary value dependent on the birank of the factor graph. Chapter 3 made progress on a topic that has never been touched before. Though the information regarding single edge addition may not seem useful because it is not recognized as a form of addition by the mathematical world, it is still useful in construction and deconstruction of graphs. If a researcher were looking at a graph and attempting to nd the biranking function, there may happen to be portion of the graph that ts this description for Single Edge

47

Addition.

This would allow the researcher to build a lower bound for one

section of the overall graph, and, with future research, see how that subgraph interacts with the other vertices of the graph. We would like to consider how arbitrary, non-vertex transitive graphs may be combined in similar ways. In Chapter 4, a topic that is commonly used to construct large graphs, the graph Cartesian product, is considered. This chapter focused on an attempt to perform initial research into the birank number of these types of graphs. Unlike single edge addition, the Cartesian Product may be a little harder to nd as a subgraph and use for a lower bound of a large, arbitrary graph. Although we would have liked to see more conclusive results such as a statement that gave a conclusive bound for at least a small number of graphs, a great push was made towards the generalization of these graphs. Future research will benet for the small portion of work done on

G3,m

graphs.

Future work would include extending Single Edge Addition to non-vertex transitive graphs. Another promising outlet would be to work towards generalizing

Cn ⊕ Cm .

general result for

Cm

Scratching the surface of this concept, it seems that the

bi(Cn ⊕ Cm ) is dependent upon the birank number of Cn

and

and also where the birank number transition are for the general formula

for cycle graphs (Theorem 5). As stated above,

G3,m

graphs seem to have a pattern that could be used

for generalization much like paths and ladders were. From there, graphs of the form

Gn,m

could be looked at and a general theorem for the birank number

may be closer then originally thought.

48

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