Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
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Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
MATH 2070 A01
H OMEWORK 9
10 March 2016
Note: Questions marked () may require some extra thought.
1. In the Knigsberg Bridge Problem, is there a route that crosses each bridge exactly twice?
2. (Ch 5, Ex 4) Use Fleurys algorithm to find an Eulerian cycle
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
MATH 2070 A01
H OMEWORK 7
25 February 2016
Solutions: Dont look until youve tried the questions!
1. Let T be a tree on n vertices with n P 3 and suppose, in hopes of a contradiction, that
all vertices in T have degree 1. Then, v2V (T ) deg(v) = n. However
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
MATH 2070 A01
H OMEWORK 8
3 March 2016
Note: Questions marked () may require some extra thought.
1. Determine the labeled tree having Prfer sequence (4, 5, 7, 2, 1, 1, 6, 6, 7).
2. The Prfer sequence of a tree T is (8, 6, 8, 8, 1, 6, 2, 2, 9). Without con
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
H OMEWORK 6
MATH 2070 A01
11 February 2016
Solutions: Dont look until youve tried the questions!
1. See the proof in your textbook.
2. Let S be any set of at most k 1 vertices and let x and y be any two vertices in G S. By
assumption, there are k internal
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
MATH 2070 A01
H OMEWORK 5
4 February 2016
Solutions: Dont look until youve tried the questions!
1. Since G is disconnected, let V (G) = A [ B be a partition of the vertex set so that there is
no edge with one endpoint in A and the other in B. Then, G cont
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
MATH 2070 A01
H OMEWORK 4
28 January 2016
Solutions: Dont look until youve tried the questions!
1. Let A [ B be a partition of V (G) with no edges within A or B. Label the vertices of G so
that A = cfw_1, 2, . . . , k and B = cfw_k + 1, k + 2, . . . , n.
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
MATH 2070 A01
H OMEWORK 7
25 February 2016
Note: Questions marked () may require some extra thought.
1. Show that every tree on at least 3 vertices contains at least one vertex of degree at least 2.
2. (Ch 3, Ex 1) Show that if T = (V, E) is a tree, then
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
MATH 2070 A01
H OMEWORK 6
11 February 2016
Note: Questions marked () may require some extra thought.
1. Prove Theorem 2.2.1 (page 50): Let G be a connected graph. Then,
(a) a vertex v is a cutvertex iff there are vertices u and w so that v is on every u
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
MATH 2070 A01
H OMEWORK 5
4 February 2016
Note: Questions marked () may require some extra thought.
1. (Ch 2, Ex 29) Let G be a disconnected graph. Show that G is connected.
2. Let G be a graph and A be its adjacency matrix. Show that G is connected iff
a
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
H OMEWORK 3
MATH 2070 A01
21 January 2016
Solutions: Dont look until youve tried the questions!
1. (a) (u, v) 7! (v, u)
(b) G: n, n2
m, G [ H: n + p, m + q, G + H: n + p, m + q + np, G H: np, nq + pm.
2. (a) (x1 , x2 , . . . , xn 1 , xn ) 7! (x1 , x2 , .
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
H OMEWORK 2
MATH 2070 A01
14 January 2016
Solutions: Dont look until youve tried the questions!
1. (a)
(b)
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
x
a1 a2 a3 a4 a5 a6
is
f (x) b1 b3 b5 b2 b4 b6
an isomorphism. To see that G1 G3 note that G3 contains the vertices c1
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
MATH 2070 A01
H OMEWORK 4
28 January 2016
Note: Questions marked () may require some extra thought.
1. (Ch 1, Ex 24) Let G be a bipartite graph. Show that the vertices of G can be ordered
and
0 M
partitioned so that, for some matrix M , the adjacency matr
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
MATH 2070 A01
H OMEWORK 3
21 January 2016
Note: Questions marked () may require some extra thought.
1. Let G be a graph with n vertices and m edges and let H be a graph with p vertices and q
edges.
(a) (Ch 1, Ex 4) Show that G H
= H G.
(b) (Ch 1, Ex 7) G
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
H OMEWORK 2
MATH 2070 A01
14 January 2016
Recall that two graphs G and H are isomorphic, denoted G
= H, iff there is a bijection
f : V (G) V (H) with the property that for every x, y V (G) cfw_x, y E(G) iff cfw_f (x), f (y)
E(H).
For example, among all
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
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Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
H OMEWORK 1
MATH 2070 A01
7 January 2016
1. For the set S = cfw_1, 2, . . . , n:
(a)
(b)
(c)
(d)
How many sets of unordered pairs are there from S?
How many ordered pairs are there from S?
How many permutations of are there of S?
How many circular permuta
Université de SaintBoniface (USB) (Affiliated with the University of Manitoba)
graph theory
MATH 2070

Winter 2016
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