The Heart of Mathematics: An Invitation to Effective Thinking

4th Edition

ISBN: 9781118156599

Author: Edward B. Burger, Michael Starbird

Publisher: Wiley, John & Sons, Incorporated

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Textbook Question

Chapter 8.4, Problem 39MS

Four parties (ExH). You want to have a party and you know 10 men and 10 women. Unfortunately, your common room can hold only 10 people. You wish to have enough parties so that each man and woman among your acquaintances would, at some point, meet at a party. How many parties are necessary to accomplish this task? Show that you need no more than four parties.

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Chapter 8.4, Problem 38MS

Chapter 8.4, Problem 40MS

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Question 4 Consider the graph G given by the following drawing. U6 U5 U4 Ա3 Աշ U1 VA V5 V6 V3 V₁ V2 (a) Determine whether G is bipartite. Show your working. (b) For each of the following sets, state whether it is a matching of G. Justify your answers. (i) {} (ii) {u2v2, U3V3, U4V4, U5V5} (iii) {u1u2, U3u4, V3 V4 V5 V6} Now consider the graph H given by the following drawing. x2 x3 x4 x5 X6 x1 Y1 Y2 Y3 Y4 Y5 Y6 (c) Find a maximum matching of H. Show your working. (d) Use Hall's theorem to show that H does not have a perfect matching.

Question 2 Let (G, w) be the network with V(G) = {a, b, c, d, e, f}, E(G) = {ab, ac, ae, af, bc, bd, bf, cd, ce, de, df, ef}, and w(ab) = 2, w(ac) = 3, w(ae) = 2, w(af) = 2, w(bc) = 3, w(bd) = 2 w(bf) = 3, w(cd) = 2, w(ce) = 1, w(de) = 2, w(df) = 2, w(ef) = 2 (a) Draw the network. (b) For each of the following trees, state whether the tree is a minimum spanning tree of (G, w). Justify your answers. (i) a b C a (iii) a b d b (c) Does there exist a minimum spanning tree of (G, w) that contains the edge af? Justify your answer. Now consider an arbitrary network (G, w), and let e = E(G). (d) Give an efficient algorithm to determine whether (G, w) has a unique minimum spanning tree. Explain briefly why the algorithm is correct and efficient.

Question 3 Consider the directed network (D, c) given by the following drawing, where each arc e = A(D) is labelled by its capacity c(e) and two vertices s and t have been identified. 3 7 3 4 5 t 3 3 8 2 (a) Give an s-t-flow of (D, c) that is not a maximum s-t-flow. Justify your answer. (b) Give an s-t-cut of (D, c) that is not a minimum s-t-cut. Justify your answer. (c) Use the Ford-Fulkerson algorithm to find a maximum s-t-flow of (D,c). Draw the residual network after each iteration of the algorithm and give the size of the maximum flow. (d) Show that the flow you have found is indeed a maximum s-t-flow.

Chapter 8 Solutions

The Heart of Mathematics: An Invitation to Effective Thinking

Ch. 8.1 - Doors galore. The 21st-century version of Lets...Ch. 8.1 - Birthday surprise. How many people would you need...Ch. 8.1 - Opposite of heads. Suppose you flip a coin 100...Ch. 8.1 - Penny percent. Suppose you flip a penny 50 times,...Ch. 8.1 - Party time. At a nephews party, you decide to...Ch. 8.1 - Flipping Lincoln. Flip a penny 100 times and...Ch. 8.1 - Flashing cards. Shuffle a standard deck of 52...Ch. 8.1 - King for a day. Remove three cards from a deck of...Ch. 8.1 - A card deal stick. Remove three cards from a deck...Ch. 8.1 - A card deal switch. Remove three cards from a deck...

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