CO342 Assignment 5 Solutions
Exercise 1:
Let G be a 3-edge-connected 3-regular graph. Prove that for any edge e in G, there is a perfect
matching of G that contains e.
Solution: Let e = xy be an arbitrary edge. To show that G has a perfect matching that c

CO342 Test 2, November 14, 11:30-12:20
1
INSTRUCTIONS
1. Fill in the following:
Name:
I.D.#:
2. No aids are allowed.
3. A complete paper has 6 pages, including this cover page. Check that you have them all.
4. You may assume the result of any theorem prov

Stable Matching
Let G = (X Y, E ) be a bipartite graph. Suppose that for each x X there
is assigned a linear ordering L(x) of the vertices in (x). This list is called the
preference list of x. Similarly for each y there is assigned a preference list L(y )

CO342 Assignment 1
Exercise 1:
Let G be a graph with n vertices such that (G) (n + 2k 4)/3. Prove that if W is a set of
vertices of G of size at most k 1 then G W has at most two components.
Exercise 2:
Let H be the graph with 4 vertices obtained by delet

CO342 Assignment 1 Solutions
Exercise 1:
Let G be a graph with n vertices such that (G) (n + 2k 4)/3. Prove that if W is a set of
vertices of G of size at most k 1 then G W has at most two components.
Solution: Let W V (G) be such that |W | k 1. Suppose o

CO342 Assignment 2 Solutions
Exercise 1:
Let G be a 2-connected graph with at least 4 vertices, and suppose C is a longest cycle in G. Let
e be an edge of C . Prove G/e is 2-connected.
Solution: Suppose on the contrary that e = xy is such that G/e is not

C&O 342 - Assignment 2
Return Date - Friday, October 7
Exercise 1:
Let G be a 2-connected graph with at least 4 vertices, and suppose C is a longest cycle in
G. Let e be an edge of C . Prove G/e is 2-connected.
Exercise 2:
Let G be a 3-regular graph. Prov

CO342 Assignment 3 Solutions
Exercise 1:
Let G be a k -connected graph with at least 2k vertices, k 2. Prove that for every set W of
vertices with |W | = k , there exists a cycle of length at least 2k in G that contains every vertex of
W.
Solution: By the

C&O 342 - Assignment 3
Return Date - Friday, October 28
Exercise 1:
Let G be a k -connected graph with at least 2k vertices, k 2. Prove that for every set W
of vertices with |W | = k , there exists a cycle of length at least 2k in G that contains every
ve

CO342 Assignment 4 Solutions
Exercise 1:
Let M be a matching in a graph G, and let V (M ) denote the set of vertices saturated by M . Prove
that there exists a maximum matching M of G that saturates V (M ).
Solution: We use induction on (G) |M |.
If (G) |

C&O 342 - Assignment 4
Return Date - Friday, November 11
Exercise 1:
Let M be a matching in a graph G, and let V (M ) denote the set of vertices saturated by
M . Prove that there exists a maximum matching M of G that saturates V (M ).
Exercise 2:
An r s L

C&O 342 - Assignment 5
Return Date - Friday, November 25
Exercise 1:
Let G be a 3-edge-connected 3-regular graph. Prove that for any edge e in G, there is a perfect
matching of G that contains e.
Exercise 2:
Is the following statement true? If so, give a

CO 342 Spring 2017: Assignment 3
Due: Wednesday May 24 at 11:00am.
1. Let k 2. Let G be a k-connected graph with at least 2k vertices. Prove that G
contains a cycle of length at least 2k.
2. Suppose G has at least 3 vertices. Prove that G is 2-connected i

CO342 Test 1, Oct 24, 11:30-12:20
1
INSTRUCTIONS
1. Fill in the following:
Name:
I.D.#:
2. No aids are allowed.
3. A complete paper has 4 pages, including this cover page. Check that you have them all.
4. You may assume the result of any theorem proved in

C&O 342 - Denition Quizzes
Sept 14: A graph consists of a set V of vertices and a set E of edges, where each edge is a subset of V of
size 2.
Sept 16: A cut-vertex of a connected graph G is a vertex x such that G x is disconnected.
Sept 19: A vertex cut o

CO 342 Final Exam, Spring 2009
Page 1
UNIVERSITY OF WATERLOO
FINAL EXAMINATION
SPRING TERM 2009
Last Name:
First Name:
Id.#:
Course Number
CO 342
Course Title
Graph Theory
Instructor
P.Haxell
Date of Exam
August 13, 2009
Time Period
7:3010:00 pm
Number of

D uwu 7D& b x q u qu x u wox zDy7P jD Durp v q u x q u |yiiojr|wz|Du x xw cfw_qu o q u w lw u u x xw x v qu uwx x xw u xo u jrDuzDjyit|xyu7jrYDrPD|vjyDytx|D1rbyyw t)|oizywVyjFu|rztVBDiz7)rvyzvjrDyqrwyDDDiz&Y xx u xqx x x xw wox cfw_ u qx o u uw xoq us x

cfw_ cfw_ x r2G $ vCDmv0$
D v
' m
@E uE
cfw_ | 'zyj bq' x x cfw_ | cfw_ z [email protected]%H%j%kC@$ukC)0 x x | cfw_ cfw_ zG$'vhz)Hz'jszk02%zklYz'5 | x |

CO 342
Assignment 10
Due: Wednesday, November 27
1. (3 points) Prove that the number of edges in a bicycle is even.
2. (5 points) Let G be a graph with no non-trivial bicycles. If e is an edge of cut type in G but
is not a cut-edge, prove that G \e contai

CO 342
Assignment 5
Due: Wednesday, October 16
1. (5 points) Show that if G is 2-connected then there is an orientation of G that is strongly
connected.
2. (5 points) Two players play a game on a connected graph by choosing a sequence of distinct
vertices

CO 342
Assignment 6
Due: Wednesday, October 23
1. (5 points) Let G be a k-connected graph, let a be a vertex in G and let B be a subset of V (G)
with |B | k. Show that there are at least k paths from a to B , with any two having only the
vertex a in commo

CO 342
Assignment 3
Due: Wednesday, October 2
1. (5 points) Prove or disprove:
(a) Any tree has at most one perfect matching.
(b) Suppose v V (G) and v has degree at least one. Then v lies in some matching of G
with maximum size.
2. (4 points) Show that a

CO 342
Assignment 9
Due: Wednesday, November 20
1. (5 points) Show that if H is a subdivision of a 2-connected graph, then it is 2-connected.
2. (5 points) Let G be a 3-connected planar graph and let C be a cycle in G with two overlapping bridges B 1 and

CO 342
Assignment 4
Due: Wednesday, October 9
1. (5 points) Let G be a 2-connected graph and let C be a cycle in G with greatest possible
length. Show that if e is an edge in C , then G/e is 2-connected.
2. (3 points) Construct a 2-connected graph G such

CO 342
Assignment 7
Due: Wednesday, November 6
1. (5 points) Let G be a connected graph with vertices u and v, and suppose that there is a
uv-separating set of size k. Prove that there are vertices a and b at distance two in G which
can be separated by a

CO 342
Assignment 7
Due: Wednesday, November 6
1. (5 points) Determine which cycles in K 3,3 and K 5 are peripheral and show that, in both
cases, each edge lies in at least three peripheral cycles.
2. (5 points) Let e be an edge in a graph G and suppose t

C&O 342
Assignment 1: corrected version
Due: 11:30am, Wednesday September 18
1. If is an automorphism of G and u V (G), show that u and (u) have
the same valency. [If S V (G), it may be useful to use (S) to denote
the set cfw_(x) : x S.]
2. Let G be a cub

CO 342
Assignment 2
Due: Wednesday, September 25
1. (5 points) List all the 2-connected simple graphs on five vertices which cannot be expressed
as non-trivial 2-sums.
2. (5 points) Show that a connected regular bipartite graph with valency at least two i

CO 342 Spring 2017 Week 5 Summary
1
Cut space
Theorem 1.1. Let G be connected and let T be a spanning tree. The set of all fundamental cuts
in G with respect to T form a basis for C(G).
Corollary 1.2. If G is connected, then dim C(G) = |V | 1.
Theorem 1.3