MA 490-505 Spring 2014
Methods of Graph Theory
A finite collection of vertices and edges is called a graph. The vertices are usually visualized as
dots, and the edges as lines which connect some or all of the pair of vertices.
The number of edges which start at a given vertex is called the degree of the vertex. A vertex
of a graph having an odd degree is called an odd vertex. A vertex having an even degree is
called an even vertex.
Handshake Lemma In any graph, the sum of the degrees of all vertices is equal to twice the
number of edges.
The number of odd vertices in any graph must be even.
Problem 1 There is a basket containing an apple, a banana, a cherry and a date. Four
children named Erica, Frank, Greg and Hank are each to be given a piece of the fruit. Erica
likes cherries and dates; Frank likes apples and cherries; Greg likes bananas and cherries; and
Hank likes apples, bananas, and dates. The problem is to give each child a piece of fruit that
he or she likes.
Problem 2 Cosmic liaisons are established among the nine planets of the solar system. Rockets travel along the following routes: Earth-Mercury, Pluto-Venus, Earth-Pluto, Pluto-Mercury,
Mercury-Venus, Uranus-Neptune, Neptune-Saturn, Saturn-Jupiter, Jupiter-Mars, and MarsUranus. Can a traveler get from Earth to Mars?
Problem 3 In the country of Figure there are nine cities, with the names 1, 2, 3, 4, 5, 6, 7, 8, 9.
A traveler finds that two cities are connected by the 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?
Problem 4 A chessboard has the form of a cross, obtained from a 4 × 4 chessboard by deleting
the corner squares. Can a knight travel this board, pass through each square exactly once, and
end on the same square he starts on?
Problem 5 Find graphs that have the following properties, if possible:
1) A graph with exactly five vertices.
2) A graph with exactly 6 edges.
3) A graph with exactly 6 edges and 7 vertices.
4) A graph with exactly 4 edges and 4 vertices.
5) A graph with exactly two odd vertices.
6) A graph with exactly three odd vertices.
7) A graph with three even vertices.
8) A graph with four even vertices.
Problem 6 In Smallville there are 15 telephones. Can they be connected by wires so that
each telephone is connected with exactly five others?
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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?
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?
Problem 9 John, coming home from Disneyland, said that he saw there an enchanted lake
with 7 island, 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 10 Can 9 line segments be drawn in the plane, each of which intersects exactly 3
others?
Problem 11 Seven student go on vacations. They decide that each will send a postcard to
three of the others. Is it possible that every student receives postcard from precisely the three
to whom he sent postcards?
Problem 12 A king has 19 vassals. Can it happen that each vassal has either 1, 5, or 9
neighbors?
Problem 13 Can a kingdom in which 3 roads lead out of each city have exactly 100 roads?
Problem 14 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.
Topics are taken from Problem-Solving Strategies by Arthur Engel, 2007; and Mathematical
Circles: Russian Experience by Dmitri Fomin, Sergey Genkin, Ilia V. Itenberg, 1996
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