graph tute 03
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The University of Sydney
School of Mathematics and Statistics
Tutorial 3 (Week 10)
MATH2069/2969: Discrete Mathematics and Graph Theory Semester 1, 2012
More difficult questions are marked with either * or **. Those marked * are at the level
which MATH2069 students will have to solve in order to be sure of getting a Credit, or to
have a chance of a Distinction or High Distinction. Those marked ** are mainly intended
for MATH2969 students.
1. For vertices v and w in a connected weighted graph, d(v, w) denotes the minimumweight of a walk from v to w. Explain why the ‘triangle inequality’
d(u, v) + d(v, w) ≥ d(u, w)
holds for any vertices u, v,w.
2. Use Dijkstra’s Algorithm to find all the minimal walks from A to Z in the followinggraph, and the weight d(A, Z ) of such a minimal walk.
1
10
6
3
10
4
2
10
1
4
1
3
6
8
2
5
3
5
2
8
5
A Z
B
C
D
E
F
G
H
I
3. Find a solution to the Chinese Postman Problem in the following graph, whereevery edge has weight 1. a b c d
e f g h
i j k ℓ
m n p q
4. Suppose that G is a connected weighted graph and u,v,w are distinct vertices.Show that any solution to the Travelling Salesman Problem for G has weight atleast d(u, v) + d(v, w) + d(w, u).
5. Draw pictures of all the isomorphism classes of trees with 5 vertices.
*6. Let T be a tree with p ≥ 2 leaves and q vertices of degree ≥ 3. Prove that q ≤ p−2.
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7. An alcohol molecule has formula C kH 2k+2O where k is a positive integer. Picturethe molecule as a connected graph where the atoms are the vertices; C atoms havedegree 4, H atoms have degree 1, and the O atom has degree 2.
(a) Show that no matter what k is, the graph is always a tree.
(b) Draw the graph for the methanol molecule, in which k = 1.
8. Find the Prufer sequences of the following trees.
(a)
6 1
42
5
3
(b)3
2
5
14
6
(c)1
2 3 4 5
6 7 8 9 10 11 12 13
9. Draw the trees with the following Prufer sequences, where the vertex set is always
{1, 2, · · · , n} for some n.(a) (1, 2, 3, 4, 5)
(b) (3, 3, 3, 3, 3)
(c) (2, 8, 6, 3, 1, 2)
10. For any positive integers m < n, let T (n, m) denote the number of trees with vertexset {1, 2, · · · , n} in which deg(n) = m.
(a) Using the bijection between trees and Prufer sequences, show that
T (n, m) =
n − 2m − 1
(n − 1)n−m−1.
(b) Hence show that for all 2 ≤ m ≤ n − 1,
(n − m)T (n, m − 1) = (n − 1)(m − 1)T (n, m).
**(c) Give a direct combinatorial proof of part (b), without using Prufer sequencesor the result of part (a). Hence give an alternative proof of part (a) and of Cayley’s Formula.
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