CHAPTER ONE: INTRODUCTION 1.1 Overview
This chapter describes a brief introduction to graph theory. History of graph theory and some famous problems are introduced. Applications of graph labelings are listed and explained to show the importance of a study in this area. Outline of this thesis with the contents of each chapter is in this chapter as well.
1.2 Brief History of Graph Theory
A research area called graph theory was started by Leonhard Euler in 1736. The citizens of Kaliningrad, Russia tried to solve if it was possible to cross all the bridges on the Pregel river only once and arrive to the starting point. Euler as a well known mathematician used graph representation and showed that it was impossible to do it.
Around 1850 another famous problem in graph theory was stated by Francis Guthrie. The problem was called the Four Color Conjecture. In this problem, a map was to be colored in such a way that all the adjacent countries sharing a border have different colors. The four color theorem was proved using a computer by Appel and Haken in 1976. A non-computer solution to the problem with an algorithm was not found until August 2004 when I. Cahit [10] proposed a non-computer proof to the problem.
In the Four Color Conjecture the vertices which showed the countries are labeled as colors.
The vertices of a graph can be labeled in different ways such as labeling the vertices with numbers. Labeling vertices and edges with numbers is a very basic and easy task. The task is more complicated when we try to include other properties in the graph as well as labeling the vertices and edges with different numbers.
The Magic-type labelings are thoroughly studied by Gallian [16]. It is stated in [16]:
“Motivated by the notation of magic squares in number theory, magic labelings were
introduced by Sedlaced in 1963.” Gallian summarized magic labelings, edge-magic total labelings, super edge-magic labelings, vertex magic total labelings and summary of antimagic labelings.
Vertex labeling methods with numbers include Vertex-Magic and Vertex-Antimagic labeling. Vertex-Magic graphs are labeled in such a way that the sum of the vertex label and its incident edge labels are same for every vertex. Vertex-Antimagic graphs are labeled so that the sum of the vertex label and its incident edge labels are different for every vertex.
Computer-assisted proofs [11] are the subject of much controversy in the mathematical world. Some mathematicians think that a computer-assisted proof is not a real mathematical proof because they involve so many logical steps that they are not verifiable by human beings, and that mathematicians are being asked to put their trust in computer programming. A reverse question can also be raised; if computer calculations are not trusted to carry out lengthy calculations, and since human beings are not infallible, why do some researchers trust in lengthy human reasoning compared to machine computation?
Other mathematicians believe that computer-assisted proofs are as valid as any other type of proof. The problem of human verifiability can be addressed by proving the computer program itself valid. The computer-assisted proofs are subject to errors in their source programs, compilers, and hardware, but this is resolved by multiple replications of the result using different programming languages, different compilers, and different computer hardware. In this thesis computer-assisted vertex-magic and vertex-antimagic total labelings are found.
1.3 Applications of Graph Labelings
Graph labelings are becoming very useful models for some applications which include coding theory problems such as the design of good radar-type codes, x-ray crystallography, communication network addressing systems and circuit designs.
Gary Bloom and Solomon Golomb [8] have researched “real world” applications of numbered undirected graphs. One of the applications to coding theory problem was to design codes for pulse radar and missile guidance. This problem is defined as labeling a graph with positive integers in such a way that the edges are different and the vertex labels determine the time positions at which pulses are transmitted.
Another application of graph labeling is in communication addressing systems. Efficient addressing systems is assigning addresses to the possible links in a communications network where the addresses all have to be different and that the addresses of a link be deduced from the identities of the two nodes linked, without having to use a lookup table.
The solution is as follows, first a graph of the network is constructed with nodes as vertices and edges between all pairs of nodes where a link is provided. The vertices of the graph are labeled in such a way that the differences between endpoints of edges are all different. Then the address of a link is the difference between the labels on its endpoints. Suppose the network graph is labeled with an edge-magic total labeling λ with a magic constant k. Then the address of the link from x to Y is immediately calculated as k x y .
Vertex-magic or vertex-antimagic labelings signify some additional information as well as identifying vertices and edges. An increase in future applications of these labelings are expected, therefore a research is done to find all possible labelings of path, cycle and tree graphs in this area.
1.4 Outline of the Thesis
The aim of this thesis is to find all possible vertex-magic and (a,d)vertex-antimagic total labelings for paths, cycles and some instances of trees. As discussed in section 1.3 the use of graph labelings in network addressing, radar pulses and many other areas is critical.
Therefore a computer program is developed to find all possible vertex-magic and (a,d)vertex-antimagic total labelings for some type of graphs. The restrictions and
boundaries of these graphs are discussed to set the domain of search area. An algorithm is designed to find all possible labelings which are included in Appendix A and Appendix B.
Preliminaries to the thesis include a brief history of graph theory, the applications of graph labelings and the outline of the thesis, in Chapter one.
In Chapter two, basic definition of graphs and basic knowledge of terms used in graph theory is explained.
Chapter three includes a brief history of vertex-magic and vertex-antimagic labelings. Basic counting on vertex-magic and vertex-antimagic labelings are also discussed in this chapter.
In Chapter four the problem to focus on in order to get all the possible results is explained.
Number of different possible labelings that is tried for vertex-magic and vertex-antimagic is explained. The relation of the number of elements in a graph with the number of tries needed to find all possible solutions is also discussed.
In Chapter five the solutions found are compared to previous proofs and open problems in this area are studied.
1.5 Summary
Graph labelings have been used for a long time and they still have applications such as producing good radar-type codes, x-ray crystallography, communication network addressing systems and circuit designs. Magic labelings signify some more additional property as well as labeling the elements of the graph. It is important to produce all possible magic labelings of graphs for further use in applications listed above.
CHAPTER TWO: GENERAL KNOWLEDGE ON GRAPHS 2.1 Overview
This chapter describes some basic definitions of graphs that will be used in this thesis.
Graph types and their properties are discussed. Vertex-magic and vertex-antimagic total labelings are explained with their formal definitions.
2.2 Basic definitions
The basic definitions of graphs and its properties are discussed by Sugeng [46] as explained; A graph is a finite set of vertices and edges where every edge connects two vertices. A graph G consists of a finite set V(G) of elements called vertices and a set E(G) of elements called edges. If x and Y are vertices in V(G) then the edge with endpoints x and Y is indicated by the xy. A graph has order v and size e, where v V(G) and e E(G) . A graph is finite if the order v is finite. A simple graph is of a kind that does not include any edge with same endpoints. All graphs discussed in this thesis are finite and simple.
Graphs are composed of nodes and lines, where nodes are called vertices and lines are called edges.
: Edge : Vertex
G1 G2 G3
Figure 2.1: Examples of simple finite graphs
u1 u2 u3
u4
v1
v2 v3
v4
v5
v6
w1 w2 w3 w
4
In a graph G, if vertex x,yV and there is an edge e between x and y, then x and Y are called adjacent vertices. Vertex x is also called the neighbour of y. Moreover, both vertices x and Y are incident with edge e. All of the neighbours of vertex x is denoted by N(x).
Degree of x is the number of neighbours of x. Therefore deg(x) N(x). If a vertex is degree 0 then it is called an isolated vertex since it has no neighbours, and a vertex with degree 1 is called an end vertex. The minimum degree of a graph G is
) deg(
min )
(G uV u
and the maximum degree is (G)maxuV deg(u). If every vertex in the graph has the same degree r, r , then G is a regular graph of degree r, or an r-regular graph.
In graph G1 vertices u1, u3 and u4 are adjacent to u2. Therefore N(u2) = {u1, u3, u4} and
3 ) (u2
N . In graph G2 since all the degrees of the vertices are the same and equal to 1, then G2 is called a 1-regular graph. Graph G1 and G3 are not regular graphs. The vertex u2 in G1 has degree 3 but the vertex u1 in the same graph has degree 1. Also the vertex w1 in G3
has degree 1 but w2 has degree 2.
Graph Y is called a subgraph of graph G if V(Y)V(G) and E(Y)E(G). The graph G is then called a supergraph of Y. A spanning subgraph Y is a subgraph of G such that
G4
Figure 2.2: Graph G4 and some of its subgraphs
u1 u
2 u3
u4
(b)
u1 u
2 u3
u4
(c)
u1 u
3
u4
(d)
u1 u
2
u4
(e)
u1 u
2 u3
u4
(f)
u1 u2 u
3
u4
(g)
u1 u2 u3
u4
) ( )
(Y V G
V . In figure 2.2 (b), (c), (d), (e), (f) and (g) are all subgraphs of G4, but only (b), (e), (f) and (g) are spanning subgraphs of G4.
A graph with n vertices x1,x2,…, xn and n-1 edges x1x2,x2x3,…,xn-1xn is called a path.
A graph with n vertices x1,x2,…, xn and n edges x1x2,x2x3,…,xn-1xn,xnx1 is called a cycle.
A graph G is connected if for any two distinct vertices u and v of G there is a path between u and v. Otherwise G is disconnected. A connected graph that does not contain a cycle is called a tree. A path is a special kind of tree. Figure 2.3 gives examples of path P5 and cycle C8.
A factor of a graph G is a spanning subgraph, a k-factor is a spanning k-regular subgraph.
Two graphs G1 and G2 of the same order are called isomorphic if there is a one-to-one mapping f from G1 to G2 that keeps the adjacency property. Thus f(v1) is adjacent to f(v2) is and only if v1 is adjacent to v2. If G1 = G2 then f is called an automorphism.
Two graphs G1 and G2 are called vertex disjoint graphs if V(G1)V(G2)0. Let G1 and G2 be two vertex disjoint graphs. A union of G1 and G2, GG1 G2, is the graph that consists of V(G)V(G1)V(G2) and E(G)E(G1)E(G2).
Figure 2.3: Path P
5 and cycle C
8
P5
v1
C8
u3 u
4 u5
v2
v3
v4
v5
v6 v7
v8 u2
u1
A complete graph Kn of order n is a graph in which every two distinct vertices are adjacent.
Let r be the degree of graph K, then r is, r n1.
2.3 General Definitions of Vertex-Magic and Vertex-Antimagic Total Labelings
Let G=(V,E) be a simple, finite and undirected graph with v vertices and e edges.
If graph G is labeled with numbers 1 through v + e such that every vertex and its incident edges adds up to the same sum for every vertex, then this labeling is called a vertex-magic total labeling of graph G. The identical sum in this graph is called the magic number. If graph G is labeled with numbers 1 through v + e such that every vertex and its incident edges add up to different sums for every vertex, then this labeling is called a vertex- antimagic total labeling of graph G.
In both vertex-magic and vertex-antimagic total labelings, the sum of all labels associated with a vertex is called the weight of that vertex. The weight of vertex xV , with labeling α, is wt(x)(x)yN(x)(xy)
In (a,d)-vertex antimagic labeling the smallest weighted vertex is a, and the other vertex weights have a constant difference of d. A labeling α : V U E {1,2,…,n+e} is called a (a,d)-vertex antimagic total labeling of G = G(V,E), if the set of vertex weights of all the vertices in G is {a, a+d,…,a+(n-1)d} where a0 and d 0 are fixed integers. If the constant difference among weights is 0, d=0, then the labeling is called vertex magic total labeling.
2.4 Summary
Graphs have some basic definitions such as vertex, edge, neighbor, subgraph etc. It is required to learn these basics about graphs to understand studies in this research area. The main focus of this thesis which is vertex-magic and vertex-antimagic total labelings are also explained with their basic definitions.
CHAPTER 3: VERTEX-MAGIC AND VERTEX-ANTIMAGIC TOTAL LABELING 3.1 Overview
In this chapter’s beginning the history of magic labelings are discussed. The relation of magic squares with magic labelings is described. Magic labelings are described, past studies about magic labelings are listed and known results are stated in this chapter in detail. Vertex-magic total labeling and vertex-antimagic total labeling is described in detail with examples and basic counting is studied.
3.2 Brief History of Magic Labeling 3.2.1 Magic Squares
Magic squares are thoroughly explained by Wallis [52]; Magic squares are among the best known mathematical recreations that have been known for ages. A magic square of side n is an nn array whose entries are an arrangement of the integers {1, 2,…, n2}, in which all elements in any row, any column, or either the main diagonal or main back-diagonal, add to the same sum as in Figure 3.1.
1 15 8 10
12 6 13 3
14 4 11 5
7 9 2 16
Figure 3.1: Magic square with side = 4
Different entries to the square are also studied, such as all entries are primes or all entries are perfect squares. Latin squares are studied since they are useful in constructing magic squares. Magic rectangles are also an area of research which can be derived from Kotzig arrays.
3.2.2 Magic Labeling
Wallis [52] explains that some authors introduced labelings that generalize the idea of a magic square. Sedlacek defined a graph to be magic if it had an edge-labeling, with range of real numbers, such that the sum of the labels around any vertex equals some constant, independent of the choice of vertex.
Kotzig and Rosa defined a magic labeling to be a total labeling in which the labels are the integers from 1 to V(G) E(G) . The sum of labels on an edge and its two endpoints is constant. In 1996 Ringel and Llado redefined this type of labeling and called it edge-magic labeling. Total labelings have also been studied in which the sum of the labels of all edges adjacent to the vertex x, plus the label of x itself, is constant.
To clarify the terminological confusion described, we define a labeling to be vertex-magic if the sum of all labels associated with a vertex equals a constant independent of the choice of a vertex, and edge-magic if the same property holds for edges. The domain of the labeling is specified by a modifier on the word “labeling”. For example, Stewart studies vertex-magic edge labelings, and Kotzig and Rosa define edge-magic total labelings. This thesis focuses on vertex-magic total labelings which is abbreviated to VMTL. The word
“total” is the modifier to the word “labeling”, in this kind of labelings all elements of the graph (edges and vertices) are labeled.
As mentioned above Sedlacek [39] introduced magic labeling in 1963. Stewart [45] studied on complete, basket and fan graphs to prove whether they can be labeled as magic graphs or not. Stewart [44] also introduced semi-magic, where the labels of edges do not need to start from 1. Jenzy and Trenkler [22] studied vertex magic edge labeling. Bodendiek and Walther [9] introduced (a,d)-vertex-antimagic edge labeling (VAE). Baca [1] also studied VAE labeling. Miller and Baca with some other researchers presented many results in magic and antimagic labelings [1, 2, 4, 5, 34, 35]. Tezer and Cahit [50] studied on paths and cycles for VAE.
Baca [3] introduced and studied (a,d)-vertex-antimagic total (VAT) labeling. Baca with some researchers have done many studies on magic and antimagic labelings that also includes VAT labelings [3,6]. MacQuillan[32] studied on various VAT labelings with different properties.
MacDougall [30, 31] introduced and studied an instance of (a,d)-VAT labeling for d=0, and he called it vertex magic total (VMT) labeling. Kovar [25,26] studied VMT labeling for regular graphs as well as studying VAT labeling for cycles.
In their paper “Vertex-magic Total Labelings of Graphs” McDougall, Miller, Slamin, Wallis [30] have studied some properties of these labelings and showed how to construct labelings for several families of graphs, including cycles, paths, complete graphs of odd order and the complete bipartite graph. They also showed that labelings are impossible for some other classes of graphs. They have proven that; the n-cycle Cn has a labeling for any
3
n . Pn, the path with n vertices, has a labeling for any n3. Every labeling of Pn is derived from a labeling if Cn.
In the study “Vertex-Magic”, Daisy Cunningham [12] has studied on bounds on magic numbers for cycles. Also, showed that if a graph has an odd number of vertices, algorithms can be found to produce different vertex-magic graphs with the maximum and minimum magic number. Cunningham has also given algorithms to produce a vertex-magic graph with odd numbers or even numbers placed on the vertices for cycle graphs. In [12] the following are also proved;
1. Let G be a cycle graph with v vertices where v is odd. There exists a vertex-magic labeling with the numbers 1 to v located on the vertices and a magic number of
2 3 27 v ,the upper bound for the magic number.
2. Let G be a cycle graph with v vertices where v is odd. There exists a vertex-magic labeling with the numbers v+1 to 2v located on the vertices and a magic number of
2 3 25 v , the lower bound for the magic number
3. Let G be a cycle graph with v vertices where v is odd. There exists a vertex-magic labeling for G with the odd numbers from 1 to 2v - 1 located on the vertices and a magic number of 3v + 2.
4. Let G be a cycle graph with v vertices where v is odd. There exists a vertex-magic labeling of G with the even numbers from 2 to 2v located on the vertices and a magic number of 3v + 1
In [12] p.19 “Some interesting questions that arise from this paper are: For a given cycle graph, is there a vertex-magic labeling associated with every magic number within the bounds?” In this thesis this open problem written above is addressed with the complete list of vertex-magic labelings produced by computer assistance.
In [3] “Vertex-antimagic Total Labelings of Graphs by Baca, Bertault, McDougall, Miller, Simanjuntak and Slamin basic properties of (a,d)-vertex antimagic total labelings (VATL) are studied. The relationships of VATL with several other previously studied graph labelings are shown. Also showed how to construct labelings for certain families of graphs.
The following proofs are summarized from this paper;
1. Every odd cycle Cn, n3, has a
2 2 ,
5
3n -vertex-antimagic total labeling and a
2 2 ,
5
5n -vertex-antimagic total labeling.
2. Every cycle Cn, n3, has a 3n2,1-vertex-antimagic total labeling and a
2n2,1-vertex-antimagic total labeling.
3. Every cycle Cn, n3, has a 3n2,2-vertex-antimagic total labeling and a
2n2,2-vertex-antimagic total labeling.
4. Every cycle Cn, n3, has a 2n2,3-vertex-antimagic total labeling and a n4,3 -vertex-antimagic total labeling.
5. Every odd cycle Cn, n3, has a n4,4-vertex-antimagic total labeling and a
n3,4-vertex-antimagic total labeling.
6. The path Pn has a (2n-1,1)-vertex-antimagic total labeling for any n2
One of the open problems for further research in [3] is as follows: “For the paths Pn and the cycles Cn, determine if there is a vertex-antimagic total labeling for every feasible pair (a,d).” In this thesis the open problem above is addressed with the complete list of
(a,d)-vertex antimagic labelings produced by computer assistance.
3.3 Description of Vertex-Magic Total Labeling
Vertex-magic total labeling is an assignment of the integers from 1 to ve to the vertices and edges of graph G so that at each vertex the vertex label and the incident edge labels add up to the same constant number.
In a more formal definition, vertex-magic total labeling is a one-to-one map λ from EV onto the integers {1,2,…, ve} if there is a constant k so that for every vertex x,
λ(x) + Σ λ(xy) = k where the sum is on all vertices Y adjacent to x. The constant k in this labeling λ is called the magic number (weight).
The notation of vertex-magic labeling was at least partially suggested by the following question which appeared on a set of mathematical enrichment problems for high school students:
The Olympic emblem consists of five overlapping rings containing 9 regions. In order to contribute to a pension fund for a retiring IOC delegate, people are asked to deposit money into each region. The guidelines allow the delegate to take all the money in any one of the rings. Place $1, $2,…,$9 in the nine regions so that the amount in each ring is the same.
The Olympic rings problem can be defined as a vertex-magic labeling of a path with 5 vertices. Therefore the solutions to vertex-magic total labeling of P5 will give the answer to the Olympic rings problem.
Weights of vertices of this graph is calculated, wt(v1) = 8 + 3 = 11
wt(v2) = 7 + 3 + 1 = 11 wt(v3) = 6 + 1 + 4 = 11 wt(v4) = 5 + 4 + 2 = 11 wt(v5) = 9 + 2 = 11
As seen above all the weights are equal. The magic constant (k) in this labeling is 11. This labeling is a vertex-magic total labeling.
8 3
6
9
1
7
4 2 5
Figure3.2: Solution to the Olympic rings problem
8 3 7 1 6 4 5 2 9
Figure3.3: Solution to the Olympic rings problem viewed in a path
v1 e1 v2 e2 v3 e3 v4 e4 v5
An example of vertex-magic total labeling of a graph is in Figure 3.4. Each vertex and its incident edges add up to 12. Every vertex-magic total labeling of a cycle graph with a magic constant k can also be labeled as edge-magic graph with same magic constant as shown in Figure 3.4. Every vertex-magic or edge-magic cycle graph can be changed to one another by just shifting each label to the next element and maintaining the order of labels.
Figure 3.4 shows just one way to create a vertex-magic graph with three vertices.
Depending upon the number of vertices and edges, a graph can be labeled in different ways with different magic numbers.
3.3.1 Basic Counting on Vertex-Magic Total Labeling
As discussed in [30]; Let M = v + e and let Sv be the sum of all vertex labels and Se the sum of all edge labels. Since the labels are from 1 to M, sum of all labels is
6
4 5
2
3
1
A vertex-magic cycle graph C
3
with a magic number of 12
1
2 3
6
4
5
An edge-magic cycle graph C
3
with a magic number of 12 Figure 3.4: Rotating the labels of a vertex-magic total label clockwise results in an edge-magic total label with same magic constant.
Sv + Se = 1Mi =
2
1 M
(1)
In a labeling λ the magic constant k is calculated by λ(x) + Σ λ(xy) = k
for only one vertex. When applied on all vertices, each vertex label is added only once and each edge label is added twice to the sum, therefore
Sv + 2Se = vk (2)
When (1) and (2) is combined,
Se +
2
1 M
= vk (3)
Vertices and edges are assigned distinct labels. Therefore the edges can receive the smallest labels 1 to e, or the largest labels v+1 to M, or anything between these two maximum and minimum points. Summarizing this we have,
e M
v
e i
S
1i
1
(4)
A similar result also holds for Sv. Combining (3) and (4) gives us,
2 1 2 2 1 2
1 2
1 vM eM vk
which will give us the range of feasible values for k. It is clear that when we know
) (G V
v and e E(G) we can find the range of k.
3.4 Description of Vertex-Antimagic Total Labeling
Vertex-antimagic total labeling is an assignment of the integers from 1 to v+e to the vertices and edges of G so that at each vertex the vertex label and the incident edge labels add up to the different numbers.
In a more formal definition, vertex-antimagic total labeling is a one-to-one map λ from E U V onto the integers {1,2,…,v+e} if the weights of vertices wt(x), x Element of V are pairwise distinct.
V U E = {1, 2,…, v + e} is called an (a,d)-vertex antimagic total labeling (VATL) of graph G if the set of vertex weights is W = { wt{x}|x ELEMENT V} = {a, a + d,…, a + (v-1)d}
for some integers a and d.
Figure 3.5 shows a vertex-antimagic total labeling and (a,d)-vertex-antimagic labeling of graph G. Let V(G) = {2, 5, 6, 8, 9} and E(G) = {1, 3, 4, 7, 10}.
3.4.1 Basic Counting on Vertex-Antimagic Total Labeling
As discussed in [3]; Let M = v + e and let Sv be the sum of all vertex labels and Se the sum of all edge labels. Since the labels are from 1 to M, sum of all labels is
8 5
6 2
9
7
10 4
1 3
wt(V1) = 2 + 1 + 3 = 6 wt(V2) = 6 + 3 + 4 = 13 wt(V3) = 5 + 4 + 7 = 16 wt(V4) = 8 + 7 + 10 = 25 wt(V5) = 9 + 10 + 1 = 20
2 8
9 6
5
10
7 4
1 3
wt(V1) = 6 + 1 + 3 = 10 wt(V2) = 9 + 3 + 4 = 16 wt(V3) = 8 + 4 + 10 = 22 wt(V4) = 2 + 10 + 7 = 19 wt(V5) = 5 + 7 + 1 = 13 Vertex-Antimagic Label (10,3)-Vertex-Antimagic Label
Figure 3.5: A Vertex-Antimagic and a (a,d)Vertex-Antimagic Total Label
Sv + Se = 1Mi =
2
1 M
(1)
Let wt(xi)aid, when summed on all vertices, each vertex label is added only once and each edge label is added twice, therefore
Sv + 2Se = (2 ( 1) )
2 a v d
v (2)
Combining (1) and (2) gives us,
Se +
2
1 M
=
v d va
2
(3)The edge labels can receive the e smallest labels or e largest labels or anything between.
Therefore we have,
M
v e
e i S i
1 1
(4) A similar result also holds for Sv. Combining (3) and (4),
2 1 2 2 1 22 1 2
1 vM v d
eM va
Shows the feasible values of a and d that are restricted. It is possible to get stronger restrictions for particular graphs.
Let δ be the smallest degree in graph G, then the minimum possible weight on a vertex is at least 1 + 2 + … + (δ + 1), therefore
2 2) 1)(
(
a (5)
Similarly if φ is the largest degree, then the maximum vertex weight is no more than the sum of φ + 1 largest labels. Therefore,
M
M i
i d
v a
) 1 (
2
) 1 )(
2
(
M
(6) Combining the inequalitites (5) and (6) the upper bound on value of d is found and shown as follows:
) 1 ( 2
) 2 )(
1 ( ) 1 )(
2 (
v
d M
(7)
3.5 Summary
Magic squares is one of the well known mathematical problems and magic labelings are introduced to generalize the idea of magic squares. Magic labelings are studied on since 1963. Vertex-magic and vertex-antimagic total labelings are discussed in detail with previously studied basic counting. It is required to study known results and previously studied basic countings to produce a good algorithm.
CHAPTER 4: ALGORITHM ON VERTEX-MAGIC AND VERTEX-ANTIMAGIC TOTAL LABELING
4.1 Overview
The problem is to find all possible vertex-magic total labelings(VMTL) and (a,d)vertex- antimagic total labelings(VATL) of cycles and paths. A computer program is written in C language to try all different possible labelings of cycles and paths. Trying all the possible labelings on a graph will give all the possible solutions of vertex-magic and vertex- antimagic total labelings.
The computer programs written are categorized in two main groups. One program is written to find VMTL and the other is for (a,d)-VATL. These programs and their algorithms are discussed in this chapter.
4.2 Program for Vertex-Magic Total Labelings
In this section the programs for cycle, path and tree graphs are observed. The problem is explained and the restrictions used in the program are discussed.
4.2.1 Vertex-Magic Total Labeling of Cycles
A program is written to find all possible vertex-magic total labelings on cycle graphs with three, four, five, six, seven and eight vertices and edges. Figure 4.1 below shows a cycle with v,e = 4, C4.
Numbers from 1 to v + e = 8 are to be labeled on Figure 4.1 to find all possible vertex- magic total labelings. All possible different labelings are labeled and checked to see if it is a vertex-magic labeling or not.
For a graph with 8 elements the number of distinct sets of V(G) and E(G) can be calculated by using combinations;
1 70 2 3 4
5 6 7 8 )!
4 8 (
! 4
! 8
4
8
C
Therefore, there are 70 different possibilities with different elements in each set in each instance. Table 4.1 shows all different combinations of labels for the cycle graph with v,e = 4, C4.
v1 v2
v3 v4
e4
e1
e2
e3
Figure 4.1: Cycle, C4, with four vertices and edges
Table 4.1: Possible distinct sets of Edges and Vertices in C4
The cycle, C4, is labeled using labeling number 3 from Table 4.1 as shown in Figure 4.2.
Realize that the weights are all different. Therefore this labeling is not a vertex-magic total labeling. After a careful observation, it is realized that by just changing the position of
No Edges Vertices No Edges Vertices No Edges Vertices
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1,2,3,4 1,2,3,5 1,2,3,6 1,2,3,7 1,2,3,8 1,2,4,5 1,2,4,6 1,2,4,7 1,2,4,8 1,2,5,6 1,2,5,7 1,2,5,8 1,2,6,7 1,2,6,8 1,2,7,8 1,3,4,5 1,3,4,6 1,3,4,7 1,3,4,8 1,3,5,6 1,3,5,7 1,3,5,8 1,3,6,7 1,3,6,8
5,6,7,8 4,6,7,8 4,5,7,8 4,5,6,8 4,5,6,7 3,6,7,8 3,5,7,8 3,5,6,8 3,5,6,7 3,4,7,8 3,4,6,8 3,4,6,7 3,4,5,8 3,4,5,7 3,4,5,6 2,6,7,8 2,5,7,8 2,5,6,8 2,5,6,7 2,4,7,8 2,4,6,8 2,4,6,7 2,4,5,8 2,4,5,7
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
1,3,7,8 1,4,5,6 1,4,5,7 1,4,5,8 1,4,6,7 1,4,6,8 1,4,7,8 1,5,6,7 1,5,6,8 1,5,7,8 1,6,7,8 2,3,4,5 2,3,4,6 2,3,4,7 2,3,4,8 2,3,5,6 2,3,5,7 2,3,5,8 2,3,6,7 2,3,6,8 2,3,7,8 2,4,5,6 2,4,5,7 2,4,5,8
2,4,5,6 2,3,7,8 2,3,6,8 2,3,6,7 2,3,5,8 2,3,5,7 2,3,5,6 2,3,4,8 2,3,4,7 2,3,4,6 2,3,4,5 1,6,7,8 1,5,7,8 1,5,6,8 1,5,6,7 1,4,7,8 1,4,6,8 1,4,6,7 1,4,5,8 1,4,5,7 1,4,5,6 1,3,7,8 1,3,6,8 1,3,6,7
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
2,4,6,7 2,4,6,8 2,4,7,8 2,5,6,7 2,5,6,8 2,5,7,8 2,6,7,8 3,4,5,6 3,4,5,7 3,4,5,8 3,4,6,7 3,4,6,8 3,4,7,8 3,5,6,7 3,5,6,8 3,5,7,8 3,6,7,8 4,5,6,7 4,5,6,8 4,5,7,8 4,6,7,8 5,6,7,8
1,3,5,8 1,3,5,7 1,3,5,6 1,3,4,8 1,3,4,7 1,3,4,6 1,3,4,5 1,2,7,8 1,2,6,8 1,2,6,7 1,2,5,8 1,2,5,7 1,2,5,6 1,2,4,8 1,2,4,7 1,2,4,6 1,2,4,5 1,2,3,8 1,2,3,7 1,2,3,6 1,2,3,5 1,2,3,4
1 5
2 3 7
8 6
4
Calculating the weights on each vertex, wt(v1) = 4 + 6 + 1 = 11
wt(v2) = 5 + 1 + 2 = 8 wt(v3) = 7 + 2 + 3 = 12 wt(v4) = 8 + 3 + 6 = 17 Figure 4.2: A labeling of C4 with calculated weights
labels in either set will result in different weights on the vertices. Without changing the elements of V and E and by just changing the positions of “1” and “3” in set E, we get Figure 4.3.
Set V(G) has four elements. It is possible to place those four elements in different positions and each position results in a different labeling. The amount of possible positions that four elements have is calculated by using permutations [38]:
24 1 2 3 4
! 4
! n
V(G) has twenty four different positions with a same set of elements. The Table 4.2 shows all the different positions of labeling number 3 from Table 4.1 with a fix positioning on E(G) and changing positions of V(G).
No Edges Vertices No Edges Vertices No Edges Vertices
1 2 3 4 5 6 7 8
1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6
4,5,7,8 4,5,8,7 4,7,5,8 4,7,8,5 4,8,5,7 4,8,7,5 5,4,7,8 5,4,8,7
9 10 11 12 13 14 15 16
1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6
5,7,4,8 5,7,8,4 5,8,4,7 5,8,7,4 7,4,5,8 7,4,8,5 7,5,4,8 7,5,8,4
17 18 19 20 21 22 23 24
1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6 1,2,3,6
7,8,4,5 7,8,5,4 8,4,5,7 8,4,7,5 8,5,4,7 8,5,7,4 8,7,4,5 8,7,5,4
Table 4.2: Elements of set V positioned in 24 different ways to produce different labelings
Labelings in Table 4.2 are drawn in Figure 4.5. Figure 4.4 shows the places of labels as a reference guide to Figure 4.5. The weights of vertices are calculated and labeled inside the gray area. The edges and vertices are labeled to the places seen in Figure 4.4.
3 5
2 1 7
8 6
4
Calculating the weights on each vertex, wt(v1) = 4 + 6 + 3 = 13
wt(v2) = 5 + 3 + 2 = 10 wt(v3) = 7 + 2 + 1 = 10 wt(v4) = 8 + 1 + 6 = 15
Figure 4.3: A different labeling of C4 with changed positions in set E
wt(V1)
V1
wt(V4)
wt(V2)
wt(V3)
V2
V3 V4
E1
E2
E3 E4
Edges = {E1,E2,E3,E4} Vertices = {V
1,V
2,V
3,V
4}
Figure 4.4: Representation of edges and vertices onto a graph
7 3 8
4 5
1 8 2 6
16 13 11
8 3 5
4 7
1 10 2 6
17 10 11
5 3 8
4 7
1 10 2 6
14 13 11
7 3 5
4 8
1 11 2 6
16 11 11
5 3 7
4 8
1 11 2 6
14 12 11
8 3 7
5 4
1 7 2 6
17 12 12
7 3 8
5 4
1 7 2 6
16 13 12
8 3 4
5 7
1 10 2 6
17 9 12
4 3 8
5 7
1 10 2 6
13 13 12
7 3 4
5 8
1 11 2 6
16 9 12
4 3 7
5 8
1 11 2 6
13 12 12
8 3 5
7 4
1 7 2 6
17 10 14
5 3 8
7 4
1 7 2 6
14 13 14
8 3 4
7 5
1 8 2 6
17 9 14
4 3 8
7 5
1 8 2 6
13 13 14
2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
8 3 7
4 5
1 8 2 6
17 12 11
1
