Ali Mahmudi, Introduction to Graph Theory [3]
The field of graph theory began to blossom in the twentieth century as more and more modeling possibilities we recognized – and growth continues. In the
mid 1800s, people began to realize that graphs could be used to model many
things that were of interest in society. For example, the “Four Color Maps Conjecture”, introduced by DeMorgan in 1852, was a famous problem that was
seemingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions
are colored differently. This conjecture can easily be phrased in terms of graph theory, and many researchers used this approach during the dozen decades that the
problem remained unsolved.
B. The Definition of Graph
In a football league there are eight teams, which we denote by A, B, C, D, E, F, G, and H. After a few weeks of the season the following games have been
played. A has played F and H
B has played E, F, and H C has played G and H
D has played E and G E has played B, D, and G
F has played A and B G has played C and E
H has played A, B, and C We may illustrate this situation by either of the two diagram of Figure 1.3,
where the teams are represented by dots and two such dots are joined by a line whenever the corresponding teams have played each other.
Figure 1.3. Games in the Football League
A B
C
D E
F G
H A
B C
D E
F G
H
Ali Mahmudi, Introduction to Graph Theory [4]
In the diagram on the left, the dots have been joined using straight lines, while in the other diagram three of lines used are not straight. But since we are
only interested in which games have been played, it does not matter whether the lines are straight or not.
The diagrams may be used to describe other situation. For example, the eight dots represent eight people, with le line joining a pair of dots if the two
people know each other. Or, the dots could be communication centre with the lines denoting communication links. Indeed, as we will see later, many real-world
situations can conveniently be described by means of such drawing, which we call graphs. A graph can be thought of as a drawing or diagram consisting of
collection of vertices dots or points together with edges lines joining certain pairs of these vertices, as in the diagrams of Figure 1.2. Here is formal definition
of graph.
If u and v are two vertices in G and e = u,v is edge in G, then i
vertex u and v are endpoints of edge e ii
vertex u and v are adjacent iii
vertex u and v incident to e or e incident to vertex u and v Consider that both graph of Figure 1.3 represent an equal graph. The vertex
set of the graph is V = {A, B, C, D, E, F, G, H} and the edge set is E = {A,F, A,H, B,E, B,F, B,H, C,G, C,H, D,E, D,G, E,G}. We can also notate
the edge set as E = {AF, AH, BE, BF, BH, CG, CH, DE, DG, EG}.
Definition 1.1
A graph G = VG, EG consist two sets, VG and VG. VG is the vertex set of the graph, often denoted by just V, which is a nonempty set of elements
called vertices. EG is the edge set of the graph, often denoted by just E, which is a possibly empty set of elements called edges, such that each edge e in E is
assigned an unordered pair of vertices u, v, called the end vertices of e.
Ali Mahmudi, Introduction to Graph Theory [5]
Example 1.1
Let G = VG, EG is a graph, where VG = {1, 2, 3, 4} and EG = {1,2, 1,3, 1,4, 2,4}. We can represent the graph as follow.
Figure 1.4. A Graph with Four Vertices and Five Edges
C. Graph as a Model
