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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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