Solution. The answer is no. We can show this by numbering the squares of the chessboard with the numbers 1, 2, 3, 9 as shown in Figure 22. Then we can represent each square by a point. If we can get from one square to another with a
39
~ ~
~ ~
FIGURE 20
~ ~
~ ~
FIGURE 21
1 4 7
2 5 8
3 6 9
FIGURE 22
knight's move, we connect the corresponding points with a line (Figure 23). The starting and ending positions of the knights are shown in Figure 24.
The order in which the knights appear on the circle clearly cannot be changed.
Therefore it is not possible to move the knights to the required positions.
The solution of these two problems, which do not resemble each other on the surface, have a central idea in common: the representation of the problem by a
5. GRAPHS-1 41
7
6 2
1 9
4 3
FIGURE 23
7
6 ~ 2 2
1"i) • 5 ~ 9 "i)9
8 8
3 3
FIGURE 24
diagram. The resulting diagrams also have something in common. Each consists of a set of points, some of which are connected by line segments.
Such a diagram is called a graph. The points are called the vertices of the graph, and the lines are called its edges.
Methodological remark. The definition we give of a graph is actually too limiting. For example, in Problem 20 below it is rather natural to draw the edges of a graph using arcs, rather than line segments. However, an accurate definition 1\'ould here be too complicated. The description above will suffice for students to get an intuitive idea of what a graDh is, which they can later refine.
* * *
Here are two more problems which can be solved by drawing graphs.
Problem 3. A chessboard has the form of a cross, obtained from a 4 x 4 chessboard by deleting the corner squares (see Figure 25). Can a knight travel around this board, pass through each square exactly once, and end on the same square he starts on?
FIGURE 25
Problem 4. In the country of Figura there are nine cities, with the names 1, 2, 3, 4, 5, 6, 7, 8, 9. A tra.veler finds that two cities are connected by an airplane route if and only if the two-digit number formed by naming one city, then the other, is divisible by 3. Can the traveler get from City 1 to City 9?
Note that one and the same graph can be represented in different ways. For example, the graph of Problem 1 can be represented as in Figure 26.
N
Ur
~Ma
FIGURE 26
The only important thing about a graph is which vertices are con- nected and which are not.
Two graphs which are actually identical, but are perhaps drawn dif- ferently, are called isomorphic.
Problem 5. Try to find. in Figures 27, 28. and 29 a graph isomorphic to the graph of Problem 2 (see Figure 23),
Solution. The first ;l.nd the third'graphs are isomorphic to each other. and it is not hard to convince oneself that both of these are isomorphic to the graph of Problem
5. GRAPHS-l 43
FIGURE 27
•
FIGURE 28
FIGURE 29
2. It suffices to renumber their vertices (see Figures 30 and 31). A proof that the graphs of Figures 28 and 23 are not isomorphic is somewhat more complicated.
For teachers. The concept of a graph should be introduced only after several problems like Problems 1 and 2 above, which involve using a graph to represent the situation of the problem. It is important that students realize right away that the
7 3
FIGURE 30
6
1
3
FIGURE 31
same graph can be drawn in different ways. To illustrate the idea of isomorphism, students cap, solve several more exercises or the type given here.
§2. The degree of a vertex: counting the edges
In the preceding section we defined a graph as a set of points (vertices), some of which are connected by lines (edges). The number of edges which start at a given
5. GRAPHS-l 45
_---.A
c
B
FIGURE 32
vertex is called the degree of the vertex. Thus, for example, in the graph of Figure 32, vertex A has degree 3, vertex B has. degree 2, and vertex C has degree l.
Problem 6. In Smallville there are 15 telephones. Can they be connected by wires so that each telephone is connected with exactly five others?
Solution. Suppose that this is possible. Consider the graph in which the vertices represent telephones, and the edges represent the wires. There are 15 vertices in this graph, and each has degree 5. Let us count the number of edges in this graph.
To do this, we can add up the degrees of all the vertices. However, in this sum, each edge is counted twice (each edge connects two vertices). Therefore the number of edges in the graph must be equal to 15ã5/2. But this number is not an integer.
It follows that such a graph cannot exist, which means that we cannot connect the telephones as required.
In solving this problem, we have shown how to count the edges of a p-aph, knowing the degree of each vertex: we add the degrees of all the vertices and divide this sum by 2.
Problem 7. In a certain kingdom, there are 100 cities, and four roads lead out of each city. How many roads are there altogether in the kingdom?
Notice that our counting method for edges of a graph has the following conse- quence: the sum of the degrees of all the vertices in a graph must be even (other- wise, we could not divide it by. 2 to get the number of edges). We can give a better formulation of this result using the following definitions:
A vertex of a graph having an odd degree is called an odd vertex. A Yertex having an even degree is called an even vertex.
Theorem. The number of odd vertices in any graph must be even.
To prove this theorem it is enough to notice that the sum of several integers is even if and only if the number of odd addends is even.
Methodological remark. This theorem plays a central role in this chapter.
It is important to keep returning to its proof, and to apply the theorem as often as possible in the solution of problems. Students should be encouraged to repeat the proof of the theorem within their solution to a problem, rather than merely quoting the theorem.
The theorem is often used to prove the existence of a certain edge of a graph, as in Problem 12. It is also used, as in Problems 8-11, to prove that a graph answering
a certain description is impossible to draw. Such problems can be difficult for students to grapple with. It i.s essential that they first try to draw the required graph, then guess that it is not possible, and finally give a clear discussion or proof, using the theorem above, that the required graph does not exist.
Problem 8. There are 30 students in a class. Can it happen that 9 of them have 3 friends each (in the class), eleven have 4 friends each, and ten have 5 friends each?
Solution. If this were possible, then it would also be possible to draw a graph with 30 vertices (representing the students), of which 9 have degree 3, 11 have degree 4, and 10 have degree 5 {by connecting "friendly" vertices with edges). However, such a graph would have 19 odd vertices, which contradicts the theorem.
Problem 9. In Smallville there are 15 telephones. Can these be connected so that (a) each telephone is connected with exactly 7 others;
(b) there are 4 telephones, each connected to 3 others, 8 telephones, each connected to 6 others, and 3 telephones, each connected to 5 others?
Problem 10. A king has 19 vassals. Can it happen that each vassal has either 1, 5, or 9 neighbors?
Problem 11. Can a kingdom in which 3 roads lead out of each city have exactly 100 roads?
Problem 12. John, coming home from Disneyland, said that he saw there an enchanted lake with 7 islands, to each of which there led either 1" 3, or 5 bridges.
Is it true that at least one of these bridges must lead to the shore of the lake?
Problem 13. Prove that the number of people who have ever lived on earth, and who have shaken hands an odd number of times in their lives, is even.
Problem 14. Can 9 line segments be drawn in the plane, each of which intersects exactly 3 oth~rs?
§3. Some new definitions
Problem 15. In the country of Seven there are 15 towns, each of which is connected to at least 7 others. Prove that one can travel from any town to any other town, possibly passing through some towns ih between.
Solution. Let us look at any 2 towns, and suppose that there is no path connecting them. This means that there is no sequence of roads such that the end of one road coincides with the beginning of the next road, connecting the 2 towns. It is given that each of the 2 towns is connected with at least 7-others. These 14 towns must be distinct: if any 2 were to coincide, there would be a path through them (or it) connecting the 2 given towns (see Figure 33). So there are at least 16 different towns, which contradicts the statement of the problem.
* * *
In -light of this problem, we give two important definitions:
A graph is called connected if any two of its vertices can be connected by a path (a sequence of edges, each of which begins at the endpoint of the previous one).
A closed path (a path whose starting and ending vertices coincide) is called a cycle.
5. GRAPHS-l 47
FIGURE 33
We can now reformulate the result of the previous problem: the graph of the roads of the kingdom of Seven is connected.
Problem 16. Prove that a graph with n vertices, each of which has degree at least (n - 1)/2, is connected.
It is natural to ask how a non-colUlected graph looks. Such a graph is composed of several "pieces", within each of which one can travel along the edges from any vertex to any other. Thus, for example, the graph of Figure 34 consists of three
"pieces" , while the graph of Figure 35 consists of two .
•
FIGURE 34
FIGURE 35
These "pieces" are called connected components of the graph. Each cOlUlected component is, of course, a connected graph. We note also that a connected graph consists of a single connected component.
Problem 17. In Never-Never-Land there is only one means of transportation:
magic .carpet. Twenty-one carpet lines servp the capital. A single line flies to Farville, and every other city is served by exactly 20 carpet lines. Show that it is possible to travel by magic carpet from the capital to Farville (perhaps by transferring from one carpet line to another).
Solution. Let us look at that connected component of the graph of carpet lines which includes the capital. We must prove that this component includes Farville.
Suppose it does not. Then there are 21 edges starting at one vertex, and 20 edges starting at every other vertex. Therefore this connected component contains exactly one odd vertex. This is a contradiction.
Methodological remark. The notion of connectedness is extremely important, and is used constantly in further work in graph theory. The important point in the solution of Problem 16-consideration of a connectedã component-is a meaningful idea, and often turns out to be useful in solving problems.
Problem 18. In a certain country, 100 roads lead out of each city, and one can travel along those roads from any city to any other. One road is closed for repairs.
Prove that one can still get from any city to any other.
§4. Eulerian graphs
Problem 19. Can one draw the graph pictured in (a) Figure 36; (b) Figure 37, without lifting the pencil from the paper, and tr"dng over each edge exactly once?
FIGURE 36
FIGURE 37
5. GRAPHS-l 49
Solution. (a) Yes. One way is to start at the vertex on the extreme left, and end at the central vertex.
(b) No. Indeed, if we can trace out the graph as required in the problem, we will arrive at every vertex as many times as we leave it (with the exception of the initial and terminal vertices). Therefore the degree of each vertex, except for two, must be even. For the graph in Figure 37 this is not the case.
In solving Problem 19, we have established the following general principle:
A graph that can be traversed without lifting the pencil from the paper, while tracing each edge exactly once, €;an have no more than two odd vertices.
This sort of graph was first studied by the great mathematician Leonhard Euler in 1736, in connection with a famous problem about the Konigsburg bridges (see also Problem 12). Graphs which can be traversed like this are called Eulerian graphs.
Problem 20. A map of the city of Konigsburg is given in Figure 38. The city lies on both banks of a river, and there are two islands in the river. There are seven bridges connecting the various parts of the city. Can one stroll around the town, crossing each bridge exactly once?
FIGURE 38
Problem 21. A group of islands are connected by bridges in such a way that one can walk from any island to any other. A tourist walked around every island, crossing each bridge exactly once. He visited the island of Thrice three times. How many bridges are there to Thrice, if
(a) the tourist neither started nor ended on Thrice;
(b) the tourist started on Thrice, but didn't end there;
(c) the tourist started and ended on Thrice?
Problem 22. (a) A piece of wire is 120 cm long. Can one use it to form the edges of a cube, each of whose edges is 10 cm?
(b) What is the smallest number of cuts one must make in the wire, so as to be able to form the required cube?
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