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 6.1, Problem 2MS
Unabridged list. Represent cach landmass from Mindscape 1 as a vertex and give each vertex a letter. Represent each bridge from Mindscape 1 as an edge of your graph using a pair of letters as in the text. Make a list of the edges. What are the degrees of each vertex of your graph?
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6.3.5 Construct a histogram for the spot weld shear strength datain Exercise 6.2.9. Comment on the shape of the histogram. Doesit convey the same information as the stem-and-leaf display?
Question 4
Consider the graph G given by the following drawing.
U6
U5
U4
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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.
Chapter 6 Solutions
The Heart of Mathematics: An Invitation to Effective Thinking
Ch. 6.1 - Map maker, map maker make me a graph. Represent...Ch. 6.1 - Unabridged list. Represent cach landmass from...Ch. 6.1 - Will the walk work? Does your graph from...Ch. 6.1 - Walk around the house. Is it possibel to traverse...Ch. 6.1 - Walk the line. Does this graph above have an Euler...Ch. 6.1 - Walkabout. Does this graph have an Euler circuit?...Ch. 6.1 - Linking the loops. In this map, the following...Ch. 6.1 - Scenic drive. (S) Here is a map of Rockystone...Ch. 6.1 - Under-edged. (H) Does this graph have an Euler...Ch. 6.1 - No man is an island. The country of Pelago...
Ch. 6.1 - Path-o-rama. For each graph below, determine if...Ch. 6.1 - Walk around the block. Create a graph of the...Ch. 6.1 - Walking the dogs. Your dogs, Abbey and Bear, love...Ch. 6.1 - Delivery query. The next time you see a postal...Ch. 6.1 - Snow job. (ExH) Shown here is a map of the tiny...Ch. 6.1 - Special delivery. (ExH) Julia is the letter...Ch. 6.1 - Draw this old house. Suppose you wanted to trace...Ch. 6.1 - Path of no return. Consider this map showing a...Ch. 6.1 - Without a trace. Is it possibel to trace out...Ch. 6.1 - New Euler. In the three previous Mindscapes, you...Ch. 6.1 - New edge—new circuit. Look at the graph for...Ch. 6.1 - New edge—new path. Review your work for...Ch. 6.1 - Path to proof. Suppose you have a connected graph...Ch. 6.1 - No Euler no how. Look at graph (a) for Mindscape...Ch. 6.1 - Degree day. (S) For cach graph below, determine...Ch. 6.1 - degrees of proof. Review your work for Mindscape...Ch. 6.1 - Degrees in sequence. Can you draw a graph that has...Ch. 6.1 - Even Steven. Review your work in Mindscape 28 to...Ch. 6.1 - Little League lesson. (H) You are in charge of...Ch. 6.1 - With a group of folks. In a small group, discuss...Ch. 6.1 - Power beyond the mathematics. Provide several...Ch. 6.1 - Here we celebrate the power of algebra as a...Ch. 6.1 - Here we celebrate the power of algebra as a...Ch. 6.1 - Here we celebrate the power of algebra as a...Ch. 6.1 - Here we celebrate the power of algebra as a...Ch. 6.1 - Here we celebrate the power of algebra as a...Ch. 6.2 - What a character! What expression gives the Euler...Ch. 6.2 - Count, then verify. What are the values of V, E,...Ch. 6.2 - Sneeze, then verify. Look at an unopened tissue...Ch. 6.2 - Blow, then verify. Inflate a ballon and use a...Ch. 6.2 - Add one. Find the values V, E, and F for the graph...Ch. 6.2 - Bowling. What is the Euler Characteristic of the...Ch. 6.2 - Making change. We begin with the graph pictured at...Ch. 6.2 - Making a point. Take a connected graph and add a...Ch. 6.2 - On the edge (H). Is it possible to add an edge to...Ch. 6.2 - Soap films. Consider the following sequence of...Ch. 6.2 - Dualing. What is the relationship between the...Ch. 6.2 - Prob. 12MSCh. 6.2 - Lots of separation. Suppose we are told that a...Ch. 6.2 - Prob. 14MSCh. 6.2 - Psychic readings. Someone is thinking of a...Ch. 6.2 - Prob. 16MSCh. 6.2 - Prob. 17MSCh. 6.2 - Circular reasoning. Create a connected graph as...Ch. 6.2 - Prob. 19MSCh. 6.2 - More circles. Consider the sphere described in...Ch. 6.2 - In the rough (S). Count the number of facets,...Ch. 6.2 - Cutting corners (H). The following collection of...Ch. 6.2 - Stellar. The following collection of pictures...Ch. 6.2 - A torus graph (ExH). The Euler Characteristic...Ch. 6.2 - Regular unfolding. Each graph below represents...Ch. 6.2 - A tale of two graphs. Suppose we draw a graph that...Ch. 6.2 - Two graph conjectures (S). Can you conjecture a...Ch. 6.2 - Lots of graphs conjecture. Can you conjecture a...Ch. 6.2 - Torus count. Three hollowed, triangular prisms...Ch. 6.2 - Torus two count (H). Carefully count the number of...Ch. 6.2 - Torus many count. Using the preceding calculations...Ch. 6.2 - Prob. 32MSCh. 6.2 - Tell the truth. Someone said that she made a...Ch. 6.2 - No sphere. Suppose we have a sphere built out of...Ch. 6.2 - Soccer ball. A soccer ball is made of pentagons...Ch. 6.2 - Klein bottle. Using the diagram here for building...Ch. 6.2 - Not many neighbors. Show that every map has at...Ch. 6.2 - Infinite edges. Suppose we consider a conn ected...Ch. 6.2 - Here we celebrate the power of algebra as a...Ch. 6.2 - Prob. 44MSCh. 6.2 - Prob. 45MSCh. 6.2 - Here we celebrate the power of algebra as a...Ch. 6.2 - Here we celebrate the power of algebra as a...Ch. 6.3 - Dont be cross. Here is a drawing of a graph with...Ch. 6.3 - De Plane! De Plane! (S) Is the graph given in...Ch. 6.3 - Countdown (H). For the graph drawing shown, count...Ch. 6.3 - Prob. 4MSCh. 6.3 - Criss-Cross. Is it possible to redraw the graph...Ch. 6.3 - Dont cross in the edge. Each of the graphs drawn...Ch. 6.3 - Hot crossed buns. Each of the graphs drawn below...Ch. 6.3 - Prob. 8MSCh. 6.3 - Spider on a mirror. Is it possible to redraw the...Ch. 6.3 - One more vertex. The graph here is drawn to show...Ch. 6.3 - Yet one more vertex (H). The graph shown is drawn...Ch. 6.3 - Familiar freckles. Is it possible to redraw the...Ch. 6.3 - Remind you of anyone you know? Is it possible to...Ch. 6.3 - Final countdown. For this graph drawing, count the...Ch. 6.3 - Euler check-up. Use your answer to the previous...Ch. 6.3 - Euler second opinion. For the graph drawing shown...Ch. 6.3 - Prob. 17MSCh. 6.3 - Prob. 18MSCh. 6.3 - A colorful museum. This figure shows the floor...Ch. 6.3 - Limit of 5. Start drawing a planar graph. Keep...Ch. 6.3 - Starring the hexagon. Is it possible to redraw...Ch. 6.3 - Prob. 22MSCh. 6.3 - Prob. 23MSCh. 6.3 - Getting greedy. (H) Suppose you are asked to color...Ch. 6.3 - Stingy rather than greedy. By coloring the...Ch. 6.3 - Getting more colorful. Graphs dont have to be...Ch. 6.3 - Prob. 27MSCh. 6.3 - Prob. 28MSCh. 6.3 - Chromatically applied. There are eight radio...Ch. 6.3 - Prob. 30MSCh. 6.3 - Personal perspectives. Write a short essay...Ch. 6.3 - Here we celebrate the power of algebra as a...Ch. 6.3 - Here we celebrate the power of algebra as a...Ch. 6.3 - Prob. 37MSCh. 6.3 - Here we celebrate the power of algebra as a...Ch. 6.3 - Here we celebrate the power of algebra as a...Ch. 6.4 - Up close and personal. Create a graph to model...Ch. 6.4 - Network lookout. Find an examle of a network...Ch. 6.4 - Prob. 3MSCh. 6.4 - Hamiltonian holiday (S). You are interning for a...Ch. 6.4 - Home style. Create a graph to model the rooms in...Ch. 6.4 - Six degrees or less. Suppose this graph is a model...Ch. 6.4 - Degrees of you. Find ten willing friends or...Ch. 6.4 - Campus shortcut. Find a map of your campus and...Ch. 6.4 - Arborist lesson. Which of the graphs below are...Ch. 6.4 - Prob. 10MSCh. 6.4 - Prob. 11MSCh. 6.4 - Prob. 12MSCh. 6.4 - Prob. 13MSCh. 6.4 - Prob. 14MSCh. 6.4 - Prob. 15MSCh. 6.4 - Hamilton Study. Look at the graph you drew to...Ch. 6.4 - Business trip redux. Look back in the section and...Ch. 6.4 - Handling Hamiltons. For each graph below, find a...Ch. 6.4 - Road trip. You are checking out gradua te programs...Ch. 6.4 - Back to Hatties trip. Look back in this section...Ch. 6.4 - Solve the Icosian Game. Find a Hamiltonian circuit...Ch. 6.4 - Hunt for Hamilton (S). A large island country has...Ch. 6.4 - Has no Hamilton. Give some characteristics that...Ch. 6.4 - Cubing Hamilton (ExH). Can you find a Hamihonian...Ch. 6.4 - Hamiltonian path. A Hamiltonian path is a path in...Ch. 6.4 - Sorry, no path. Give some characteristics that...Ch. 6.4 - Prob. 27MSCh. 6.4 - Prob. 28MSCh. 6.4 - Prob. 29MSCh. 6.4 - Prob. 30MSCh. 6.4 - Edge count. Look at all the trees you drew in the...Ch. 6.4 - Personal perspecthes. Write a short essay...Ch. 6.4 - Prob. 33MSCh. 6.4 - Prob. 34MSCh. 6.4 - Dollars and cents. Your spanning tree has three...Ch. 6.4 - Adding up. Your spanning tree has four edges with...Ch. 6.4 - Prob. 38MSCh. 6.4 - Vertex search (H). Your graph has a Hamiltonian...Ch. 6.4 - Binary gossip tree. You told a secret to two of...
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