MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 1 solution
Question 1.
(1 point) Exercise 1.1.9. Prove that the graph on the left is isomorphic to the
complement of the graph on the right.
1
a
1
0
1
0
2
5
6
1
0

MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 3 solution
Question 1. (2 points) Exercise 1.4.29. Suppose that G is a graph and D is a strong orientation
of G.
(a) Consider an edge cfw_x, y in G. Prove that i

MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 2 solution
Question 1.
(2 points) (a) A graph G is dened to be connected if every pair of vertices is
connected by a path. Let G be a graph and x V (G). Prove tha

MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 4 solution
Question 1. (2 points) (a) Describe the breadth rst search (BFS) algorithm when applied to a
connected graph.
(b) Explain how BFS can be used to nd the

MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 5 solution
Question 1.
(1 point) Exercise 3.1.8. Prove that every tree T contains at most one perfect
matching. (Hint. If M and M are two perfect matchings in T ,

MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 8 solution
Question 1. Exercises 7.2.4 and 7.2.7. Note that a Hamiltonian path in a graph G is a spanning
path of G.
(a) Prove that if G has a Hamiltonian path th

MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 7 solution
Question 1. (1 point) Exercise 5.1.12, 5.1.13, 5.1.14, 5.1.15.
(a) Let (G) denote the independence number of of a graph G. Prove or disprove: If G is a

MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 6 solution
Question 1. (2 points) Exercise 3.3.16. Let G be a k -regular graph of even order that remains
connected when any k 2 edges are removed. We shall follo