640 Chapter 10 Graphs and Trees For each of the graphs in 8 and 9: (i) Find all edges that are incident on v1 . (ii) Find all vertices that are adjacent to v3 . (iii) Find all edges that are adjacent to e1 . (iv) Find all loops. (v) Find all parallel edges. (vi) Find all isolated vertices. (vii) Find the degree of v3 . (viii) Find the total degree of the graph. 8.
v1
In each of 17–25, either draw a graph with the specified properties or explain why no such graph exists. 17. Graph with five vertices of degrees 1, 2, 3, 3, and 5. 18. Graph with four vertices of degrees 1, 2, 3, and 3. 19. Graph with four vertices of degrees 1, 1, 1, and 4. 20. Graph with four vertices of degrees 1, 2, 3, and 4.
e6 e2
e8
v6
e5
e4
e 10
22. Simple graph with five vertices of degrees 2, 3, 3, 3, and 5. 23. Simple graph with five vertices of degrees 1, 1, 1, 2, and 3.
v3
24. Simple graph with six edges and all vertices of degree 3.
e7
v4
e9
21. Simple graph with four vertices of degrees 1, 2, 3, and 4.
v2
e3
e1
v5
16. A graph has vertices of degrees 1, 1, 4, 4, and 6. How many edges does the graph have?
25. Simple graph with nine edges and all vertices of degree 3.
9.
e3
e1 e2 v1
a. e6
e4 v5
26. Find all subgraphs of each of the following graphs.
v2 e7
e5
v3
v4
10. Use the graph of Example 10.1.6 to determine a. whether Sports Illustrated contains printed writing; b. whether Poetry Magazine contains long words. 11. Find three other winning sequences of moves for the vegetarians and the cannibals in Example 10.1.7. 12. Another famous puzzle used as an example in the study of artificial intelligence seems first to have appeared in a collection of problems, Problems for the Quickening of the Mind, which was compiled about A.D. 775. It involves a wolf, a goat, a bag of cabbage, and a ferryman. From an initial position on the left bank of a river, the ferryman is to transport the wolf, the goat, and the cabbage to the right bank. The difficulty is that the ferryman’s boat is only big enough for him to transport one object at a time, other than himself. Yet, for obvious reasons, the wolf cannot be left alone with the goat, and the goat cannot be left alone with the cabbage. How should the ferryman proceed? 13. Solve the vegetarians-and-cannibals puzzle for the case where there are three vegetarians and three cannibals to be transported from one side of a river to the other. H 14. Two jugs A and B have capacities of 3 quarts and 5 quarts, respectively. Can you use the jugs to measure out exactly 1 quart of water, while obeying the following restrictions? You may fill either jug to capacity from a water tap; you may empty the contents of either jug into a drain; and you may pour water from either jug into the other. 15. A graph has vertices of degrees 0, 2, 2, 3, and 9. How many edges does the graph have?
v2
b.
e1
c.
v2
v1 e2
v1
v0
v1
v3
27. a. In a group of 15 people, is it possible for each person to have exactly 3 friends? Explain. (Assume that friendship is a symmetric relationship: If x is a friend of y, then y is a friend of x.) b. In a group of 4 people, is it possible for each person to have exactly 3 friends? Why? 28. In a group of 25 people, is it possible for each to shake hands with exactly 3 other people? Explain. 29. Is there a simple graph, each of whose vertices has even degree? Explain. 30. Suppose that G is a graph with v vertices and e edges and that the degree of each vertex is at least dmin and at most dmax . Show that 1 1 dmin · v ≤ e ≤ dmax · v. 2 2 31. Prove that any sum of an odd number of odd integers is odd. H 32. Deduce from exercise 31 that for any positive integer n, if there is a sum of n odd integers that is even, then n is even. 33. Recall that K n denotes a complete graph on n vertices. a. Draw K 6 . H b. Show that for all integers n ≥ 1, the number of edges of n(n − 1) . K n is 2 34. Use the result of exercise 33 to show that the number of edges of a simple graph with n vertices is less than or equal n(n − 1) to . 2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
640 Chapter 10 Graphs and Trees For each of the graphs in 8 and 9: (i) Find all edges that are incident on v1 . (ii) Find all vertices that are adjacent to v3 . (iii) Find all edges that are adjacent to e1 . (iv) Find all loops. (v) Find all parallel edges. (vi) Find all isolated vertices. (vii) Find the degree of v3 . (viii) Find the total degree of the graph. 8.
v1
In each of 17–25, either draw a graph with the specified properties or explain why no such graph exists. 17. Graph with five vertices of degrees 1, 2, 3, 3, and 5. 18. Graph with four vertices of degrees 1, 2, 3, and 3. 19. Graph with four vertices of degrees 1, 1, 1, and 4. 20. Graph with four vertices of degrees 1, 2, 3, and 4.
e6 e2
e8
v6
e5
e4
e 10
22. Simple graph with five vertices of degrees 2, 3, 3, 3, and 5. 23. Simple graph with five vertices of degrees 1, 1, 1, 2, and 3.
v3
24. Simple graph with six edges and all vertices of degree 3.
e7
v4
e9
21. Simple graph with four vertices of degrees 1, 2, 3, and 4.
v2
e3
e1
v5
16. A graph has vertices of degrees 1, 1, 4, 4, and 6. How many edges does the graph have?
25. Simple graph with nine edges and all vertices of degree 3.
9.
e3
e1 e2 v1
a. e6
e4 v5
26. Find all subgraphs of each of the following graphs.
v2 e7
e5
v3
v4
10. Use the graph of Example 10.1.6 to determine a. whether Sports Illustrated contains printed writing; b. whether Poetry Magazine contains long words. 11. Find three other winning sequences of moves for the vegetarians and the cannibals in Example 10.1.7. 12. Another famous puzzle used as an example in the study of artificial intelligence seems first to have appeared in a collection of problems, Problems for the Quickening of the Mind, which was compiled about A.D. 775. It involves a wolf, a goat, a bag of cabbage, and a ferryman. From an initial position on the left bank of a river, the ferryman is to transport the wolf, the goat, and the cabbage to the right bank. The difficulty is that the ferryman’s boat is only big enough for him to transport one object at a time, other than himself. Yet, for obvious reasons, the wolf cannot be left alone with the goat, and the goat cannot be left alone with the cabbage. How should the ferryman proceed? 13. Solve the vegetarians-and-cannibals puzzle for the case where there are three vegetarians and three cannibals to be transported from one side of a river to the other. H 14. Two jugs A and B have capacities of 3 quarts and 5 quarts, respectively. Can you use the jugs to measure out exactly 1 quart of water, while obeying the following restrictions? You may fill either jug to capacity from a water tap; you may empty the contents of either jug into a drain; and you may pour water from either jug into the other. 15. A graph has vertices of degrees 0, 2, 2, 3, and 9. How many edges does the graph have?
v2
b.
e1
c.
v2
v1 e2
v1
v0
v1
v3
27. a. In a group of 15 people, is it possible for each person to have exactly 3 friends? Explain. (Assume that friendship is a symmetric relationship: If x is a friend of y, then y is a friend of x.) b. In a group of 4 people, is it possible for each person to have exactly 3 friends? Why? 28. In a group of 25 people, is it possible for each to shake hands with exactly 3 other people? Explain. 29. Is there a simple graph, each of whose vertices has even degree? Explain. 30. Suppose that G is a graph with v vertices and e edges and that the degree of each vertex is at least dmin and at most dmax . Show that 1 1 dmin · v ≤ e ≤ dmax · v. 2 2 31. Prove that any sum of an odd number of odd integers is odd. H 32. Deduce from exercise 31 that for any positive integer n, if there is a sum of n odd integers that is even, then n is even. 33. Recall that K n denotes a complete graph on n vertices. a. Draw K 6 . H b. Show that for all integers n ≥ 1, the number of edges of n(n − 1) . K n is 2 34. Use the result of exercise 33 to show that the number of edges of a simple graph with n vertices is less than or equal n(n − 1) to . 2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
642 Chapter 10 Graphs and Trees 46. Imagine that the diagram shown below is a map with countries labeled a–g. Is it possible to color the map with only three colors so that no two adjacent countries have the same color? To answer this question, draw and analyze a graph in which each country is represented by a vertex and two vertices are connected by an edge if, and only if, the countries share a common border.
b
a d
c e
f
g
H 47. In this exercise a graph is used to help solve a scheduling problem. Twelve faculty members in a mathematics department serve on the following committees:
a schedule that will allow all faculty members to attend the meetings of all committees on which they serve. To do this, represent each committee as the vertex of a graph, and draw an edge between two vertices if the two committees have a common member. Find a way to color the vertices using only three colors so that no two committees have the same color, and explain how to use the result to schedule the meetings. 48. A department wants to schedule final exams so that no student has more than one exam on any given day. The vertices of the graph below show the courses that are being taken by more than one student, with an edge connecting two vertices if there is a student in both courses. Find a way to color the vertices of the graph with only four colors so that no two adjacent vertices have the same color and explain how to use the result to schedule the final exams.
Undergraduate Education: Tenner, Peterson, Kashina, Cohen
MCS101
Graduate Education: Gatto, Yang, Cohen, Catoiu Colloquium: Sahin, McMurry, Ash Library: Cortzen, Tenner, Sahin
MCS102
MCS100
MCS110
Hiring: Gatto, McMurry, Yang, Peterson Personnel: Yang, Wang, Cortzen The committees must all meet during the first week of classes, but there are only three time slots available. Find
MCS135 MCS130
MCS120
Answers for Test Yourself 1. a finite, nonempty set of vertices; a finite set of edges; one or two vertices called its endpoints 2. an edge with a single endpoint 3. they have the same set of endpoints 4. they are connected by an edge 5. each of its endpoints 6. adjacent 7. isolated 8. an ordered pair of vertices called its endpoints 9. a graph with no loops or parallel edges 10. simple graph with n vertices whose set of edges contains exactly one edge for each pair of vertices 11. connected by an edge; any other vertex in V1 ; any other vertex in V2 12. every vertex in H is also a vertex in G; every edge in H is also an edge in G; every edge in H has the same endpoints as it has in G 13. the number of edges that are incident on the vertex, with an edge that is a loop counted twice 14. the sum of the degrees of all the vertices of the graph 15. equal to twice the number of edges of the graph 16. an even number
10.2 Trails, Paths, and Circuits One can begin to reason only when a clear picture has been formed in the imagination. — W. W. Sawyer, Mathematician’s Delight, 1943
The subject of graph theory began in the year 1736 when the great mathematician Leonhard Euler published a paper giving the solution to the following puzzle: The town of Königsberg in Prussia (now Kaliningrad in Russia) was built at a point where two branches of the Pregel River came together. It consisted of an island and some land along the river banks. These were connected by seven bridges as shown in Figure 10.2.1. The question is this: Is it possible for a person to take a walk around town, starting and ending at the same location and crossing each of the seven bridges exactly once?∗ ∗
In his original paper, Euler did not require the walk to start and end at the same point. The analysis of the problem is simplified, however, by adding this condition. Later in the section, we discuss walks that start and end at different points.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
658 Chapter 10 Graphs and Trees 8. Find the number of connected components for each of the following graphs. a.
b
c
a
15.
t
b
i
g
e
a
14.
v u
s
w
h
f
h
g
f
d
r
c
e
z
y
x
d
v
b. u
w
z
x
16.
y
c.
d. g
c
b
i
a
e
j
v0
v2
v5
v3
v2 v3 v4
f
h
11. Is it possible for a citizen of Königsberg to make a tour of the city and cross each bridge exactly twice? (See Figure 10.2.1.) Why? Determine which of the graphs in 12–17 have Euler circuits. If the graph does not have an Euler circuit, explain why not. If it does have an Euler circuit, describe one. 13.
e1 v1 e8
e2
e3
e7
v5 e6
v4
E
F
D River
E
For each of the graphs in 19–21, determine whether there is an Euler path from u to w. If there is, find such a path. v0
19. v7
20. v1
b
c
a
u
d
u
e4
v6
v1 v2
v8 v3
D
C
A
v0
v9
C
B
10. The solution for Example 10.2.5 shows a graph for which every vertex has even degree but which does not have an Euler circuit. Give another example of a graph satisfying these properties.
v2
B
18. Is it possible to take a walk around the city whose map is shown below, starting and ending at the same point and crossing each bridge exactly once? If so, how can this be done?
9. Each of (a)–(c) describes a graph. In each case answer yes, no, or not necessarily to this question: Does the graph have an Euler circuit? Justify your answers. a. G is a connected graph with five vertices of degrees 2, 2, 3, 3, and 4. b. G is a connected graph with five vertices of degrees 2, 2, 4, 4, and 6. c. G is a graph with five vertices of degrees 2, 2, 4, 4, and 6.
12.
17. A
v4
v1
d
v1
v2
v5 v5
e5
v3 v4
v7
e
f h
w
v3 w
v4
g
v6
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
10.7
Spanning Trees and Shortest Paths
715
Exercise Set 10.7 Find all possible spanning trees for each of the graphs in 1 and 2. 1.
a
d
c
v0
w
v3
v2
b
d
c
x
c
7
g
10
e
9
2
6
4 3 5
7
4 v5
v4 8
5 v3
15
18
13
v6
4 5
14.
8. The graph of exercise 6.
g
b 7
4 1
f
e
8 z
1 2
d
e
10
19
For each of the graphs in 9 and 10, find all minimum spanning trees that can be obtained using (a) Kruskal’s algorithm and (b) Prim’s algorithm starting with vertex a or t. Indicate the order in which edges are added to form each tree.
4
c
5 1
v2
7. The graph of exercise 5.
3
3
b a
Use Prim’s algorithm starting with vertex a or v0 to find a minimum spanning tree for each of the graphs in 7 and 8. Indicate the order in which edges are added to form each tree.
a
12. Use Dijkstra’s algorithm for the airline route system of Figure 10.7.3 to find the shortest distance from Nashville to Minneapolis. Make a table similar to Table 10.7.1 to show the action of the algorithm.
20
v7
3
Amarillo
2
2
10
Phoenix
13.
v1
12
1.1 Albuquerque
Use Dijkstra’s algorithm to find the shortest path from a to z for each of the graphs in 13–16. In each case make tables similar to Table 10.7.1 to show the action of the algorithm.
d
f
v0
2.2
2.9 1.2
1 8
1.8 1.7
v
w
Use Kruskal’s algorithm to find a minimum spanning tree for each of the graphs in 5 and 6. Indicate the order in which edges are added to form each tree.
a
0.8
2.5
x y
b
Cheyenne
1.5 1.6
1.9
u
5.
z
5
Salt Lake City
t
z
9.
y
2
7
10
Denver
s
r
v
8
1
f
4.
6.
2
11. A pipeline is to be built that will link six cities. The cost (in hundreds of millions of dollars) of constructing each potential link depends on distance and terrain and is shown in the weighted graph below. Find a system of pipelines to connect all the cities and yet minimize the total cost.
e
g
5
u
7
v1
Find a spanning tree for each of the graphs in 3 and 4. 3. a
t 3 15
2.
b
10.
7 11 12
c 10 d
b
1
c
1
d 20
1 7
a 4
8
z 1
e 1
f
1
g
15. The graph of exercise 9 with a = a and z = f 16. The graph of exercise 10 with a = u and z = w 17. Prove part (2) of Proposition 10.7.1: Any two spanning trees for a graph have the same number of edges. 18. Given any two distinct vertices of a tree, there exists a unique path from one to the other. a. Give an informal justification for the above statement. ✶ b. Write a formal proof of the above statement.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
CHAPTER 8
6
1
5
â–
Basic Graph Theory
1
2
3
4
5
6
493
2
4
3
Show that the power grids are equivalent and, hence, that Harvard can employ the same diagnostic software as that used at MIT. 22. Show that Qn is vertex transitive. 23. Are the graphs in Exercise 21 vertex transitive? 24. Is the pictured graph isomorphic to the Petersen graph?
25. Show that there is a graph map from C4 to the pictured graph. b a c
26. How many automorphisms does the graph pictured in Exercise 25 have? !27. Find |Aut( Q3 )|. Compare this result with the number of automorphisms of a die studied in Example 7.18 of Section 7.4. 28. Is there a graph map from the tetrahedron to the octahedron? Explain. 29. Show that, if G = (V, E) is an r -regular graph and r is odd, then |V| is even. 30. Show that the two pictured graphs are not isomorphic.
31. Pictured are two design plans for a power grid at Yale, in which the symbol ! denotes a power station. They seem to have one major design difference around the interior power station on the right. Are they really different?
Show that the two graphs that model these grids are in fact not isomorphic.
CHAPTER 8
â–
Basic Graph Theory
495
42. Is the pictured directed graph simple?
43. Pictured are two seemingly different designs for a computer network on six computers using eight cables in which each cable connection allows message passing in only one direction. 1
2
3
7
8
9
4
5
6
10
11
12
Show that the two directed graphs that model these networks are isomorphic. 44. Show that the two pictured directed graphs are not isomorphic. 1
2
1
4
3
2
4
3
45. Determine whether the two pictured directed graphs are isomorphic. 1
2
3
4
5
1
2
3
4
46. List all non-isomorphic simple directed graphs on two vertices. 47. List all directed graphs G for which G is the pictured graph. 1
2
3
4
5
506
P A R T II
■
Combinatorics
37. Consider the power grid from Exercise 13. Determine the edge connectivity of the graph that models this grid, and prove your result. Is there a unique λ-set? 38. Consider the computer network from Exercise 14. Determine the edge connectivity of the graph that models this network, and prove your result. Is there a unique λ-set? 39. An edge in a graph whose removal disconnects the graph is called a cut-edge or bridge. The presence of such an edge strongly ties the connectivity of the graph to its edge connectivity. Let G be any graph. Show: If λ(G) ≤ 1, then κ(G) = λ(G).
40. We saw in Theorem 9.6 that 3-regular graphs have edge connectivity equal to their vertex connectivity. Prove or disprove: If G is any 4-regular graph, then λ(G) = κ(G). 41. Show: λ( K n ) = n − 1.
42. Prove the assertions in Example 9.4. 43. Give an example of a simple graph G for which κ(G) < λ(G) = δ(G).
44. Give an example of a simple graph G for which κ(G) = λ(G) < δ(G). In Exercises 45 through 50, determine (a) the connectivity κ and (b) the edge connectivity λ of the specified directed graph G = (V, E). 45. V = {1, 2, 3, 4, 5} and E = {(1, 3), (5, 1), (2, 3), (4, 2), (3, 4), (3, 5)}. 46. V = {1, 2, 3, 4, 5, 6} and E = {(1, 3), (2, 1), (3, 2), (4, 3), (4, 6), (5, 4), (6, 1), (6, 5)}. 47. V = {1, 2, 3, 4} and E = {(1, 3), (2, 1), (3, 2), (3, 4), (4, 1), (2, 2)}. 48. V = {1, 2, 3, 4} and E = {(1, 2), (1, 3), (1, 4), (2, 3), (4, 3)}.
49. In the pictured neighborhood, the portions of the streets marked with arrows are one-way streets, whereas the others are two-way. Let G be the directed graph that reflects the resulting traffic flow.
50. In the pictured neighborhood, the portions of the streets marked with arrows are one-way streets, whereas the others are two-way. Let G be the directed graph that reflects the resulting traffic flow.
CHAPTER 9
â&#x2013;
Graph Properties
513
Nord 9
8
5 4 3
6
7
Seine Ile de la CitĂŠ
15
Ile St-Louis
10 11 12 13 14
1 2
Sud
With its many bridges, this section might appeal to tourists wanting to cross over each bridge. Determine whether this portion of Paris has an Euler circuit, an Euler trail, or neither. 12. Pictured is a portion of Rome through which the Tiber River flows. Teatre Marcello 3
Tiber
1
Tiberina 2 4
1 ! Ponte Garibaldi 2 ! Ponte Crestio 3 ! Ponte Fabricio 4 ! Ponte Palatino
For a tourist interested in its bridges, determine whether this portion of Rome has an Euler circuit, an Euler trail, or neither. 13. A street sweeper needs to sweep each side of the two-way streets in the pictured neighborhood.
To be maximally efficient, the route should cover each side of each street exactly once. Find a maximally efficient route through this neighborhood for the street sweeper. 14. To protect against cracking pavement, a sealant needs to be applied to the surfaces of the streets in the pictured neighborhood.
514
P A R T II
â&#x2013;
Combinatorics
With one pass, a truck can spray the sealant on one side of these two-way streets. Find a route through the neighborhood for the sealant truck so that it covers each side of each street exactly once. 15. Suppose a graph G has an Euler trail and is not Eulerian. Is it possible to add a single edge to G to make the resulting graph Eulerian? Explain. 16. Suppose a graph G has an Euler trail and the two vertices of odd degree are endpoints of an edge e. Must G\{e} be Eulerian? Explain. 17. Suppose a graph G has c components and exactly two vertices u and v of odd degree. Can u and v be in different components? What is the minimum number of edges that must be added to G to obtain a graph with an Euler trail? 18. Suppose a graph G has c components and each vertex has even degree. What is the minimum number of edges that must be added to G to obtain an Eulerian graph? Does it matter where the edges are attached? 19. A standard assignment for a janitor is to mop and wax the hallways in some building. Of course, each floor area should be mopped first and waxed second. Also, it is preferred that the janitor not walk on a floor once it has been waxed. Consequently, the janitor should enter the building, mop and wax all of the floors, and then exit the building, leaving the floors to dry overnight. Prove that, no matter what the layout of the hallways in the building, the janitor can always find a plan that accomplishes this. 20. Once a building has been constructed, its walls need to be painted. Suppose one man is assigned the job of painting the walls of all of the hallways in a single color. Since the ladder and other materials are cumbersome to move around, the man would like to complete the job in one continuous route around the building, a route that does not require him to carry his materials down a hallway both of whose walls have already been painted. Prove that, no matter what the layout of the hallways in the building, the painter can always find a plan that accomplishes this. !21. Prove that the equivalent conditions in Theorem 9.7(a) are also equivalent to the condition that the edges of the graph G can be split into a disjoint union of cycles. 22. Prove Theorem 9.7(b). 23. For which integers n is Qn Eulerian? 24. For which integers n is K n Eulerian? 25. For which pairs of integers m, n is K m,n Eulerian? 26. Let G = (V, E) be a graph with |V| odd such that both G and G c are connected. Show: G is Eulerian if and only if G c is Eulerian. 27. What is the minimum possible number of edge repetitions in a circuit that covers every edge of K 2,3 ? 28. What is the minimum possible number of edge repetitions in a circuit that covers every edge of K 4 ? 29. Consider the layout of a portion of Paris shown in Exercise 11. A tourist desires a walk that crosses every bridge and returns to its starting point. (a) What is the smallest possible number of bridges that would need to be crossed more than once in order for the tourist to cross every bridge?
514
P A R T II
â&#x2013;
Combinatorics
With one pass, a truck can spray the sealant on one side of these two-way streets. Find a route through the neighborhood for the sealant truck so that it covers each side of each street exactly once. 15. Suppose a graph G has an Euler trail and is not Eulerian. Is it possible to add a single edge to G to make the resulting graph Eulerian? Explain. 16. Suppose a graph G has an Euler trail and the two vertices of odd degree are endpoints of an edge e. Must G\{e} be Eulerian? Explain. 17. Suppose a graph G has c components and exactly two vertices u and v of odd degree. Can u and v be in different components? What is the minimum number of edges that must be added to G to obtain a graph with an Euler trail? 18. Suppose a graph G has c components and each vertex has even degree. What is the minimum number of edges that must be added to G to obtain an Eulerian graph? Does it matter where the edges are attached? 19. A standard assignment for a janitor is to mop and wax the hallways in some building. Of course, each floor area should be mopped first and waxed second. Also, it is preferred that the janitor not walk on a floor once it has been waxed. Consequently, the janitor should enter the building, mop and wax all of the floors, and then exit the building, leaving the floors to dry overnight. Prove that, no matter what the layout of the hallways in the building, the janitor can always find a plan that accomplishes this. 20. Once a building has been constructed, its walls need to be painted. Suppose one man is assigned the job of painting the walls of all of the hallways in a single color. Since the ladder and other materials are cumbersome to move around, the man would like to complete the job in one continuous route around the building, a route that does not require him to carry his materials down a hallway both of whose walls have already been painted. Prove that, no matter what the layout of the hallways in the building, the painter can always find a plan that accomplishes this. !21. Prove that the equivalent conditions in Theorem 9.7(a) are also equivalent to the condition that the edges of the graph G can be split into a disjoint union of cycles. 22. Prove Theorem 9.7(b). 23. For which integers n is Qn Eulerian? 24. For which integers n is K n Eulerian? 25. For which pairs of integers m, n is K m,n Eulerian? 26. Let G = (V, E) be a graph with |V| odd such that both G and G c are connected. Show: G is Eulerian if and only if G c is Eulerian. 27. What is the minimum possible number of edge repetitions in a circuit that covers every edge of K 2,3 ? 28. What is the minimum possible number of edge repetitions in a circuit that covers every edge of K 4 ? 29. Consider the layout of a portion of Paris shown in Exercise 11. A tourist desires a walk that crosses every bridge and returns to its starting point. (a) What is the smallest possible number of bridges that would need to be crossed more than once in order for the tourist to cross every bridge?
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A graph G is planar if and only if ν(G) = 0. Since K 5 and K 3,3 are not planar and Examples 9.18 and 9.19 show drawings of them with one crossing, it follows that ν( K 5 ) = ν( K 3,3 ) = 1. In applications such as network or circuit design, when the number of wire crossings must be minimized, an embedding that achieves the crossing number of the corresponding graph is sought. EXAMPLE 9.25
The layout for the power grid in Example 9.24 has crossing number 1. That the crossing number is at least 1 is established in Example 9.24 by showing that the graph modeling this grid is not planar. That the grid can be laid out with just one crossing is shown in the following picture. 1 8
2
5
4
6 3
7
SECTION 9.4
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Exercises 1.
2.
3.
Show that the pictured computer network configuration can be laid out without cable crossings. 1
2
3
4
5
Pictured is a design plan for a power grid, in which the symbol ◦ denotes a power station. Show that there is a layout for this power grid without cable crossings. 1
2
4
3
5
When connecting the components of a personal computer, one typically ends up with a mess of tangled cables stuffed behind the computer. Show that the connections in the pictured computer system can be accomplished without crossings.
CHAPTER 9
Tower
â&#x2013;
Graph Properties
535
Monitor
Printer Speakers
Keyboard
Mouse Power Strip
4.
A camp is being built to contain three buildings. There are four utilities available, but it is not necessary for each building to be connected to each utility. Show that the required connections shown in the diagram below can be accommodated without cable crossings. Gas Electric Water
Bathrooms Sleeping Quarters Mess Hall
Sewer
5.
Show that K 2,3 is planar by giving a drawing without crossings.
6.
Two-Utility Problem. If there are just 2 utilities and any number n of homes, then each home can be connected to each utility without any cable crossings. Show that, for any n â&#x2C6;&#x2C6; Z+ , K 2,n is planar.
7.
Draw the dual of the pictured embedding of a graph.
9.
Exercises 15 and 16 from Section 8.3 show that the cube and the octahedron are planar graphs. Show that the dual of the cube is the octahedron, and vice versa.
8.
Draw the dual of the pictured embedding of a graph.
10. Exercises 14 and 18 from Section 8.3 show that the dodecahedron and the icosahedron are planar graphs. Show that the dual of the dodecahedron graph is the icosahedron graph, and vice versa. 11. Given a planar embedding of a graph G = (V, E) with regions R, if |V| = 10 and |E| = 16, then find |R|. 12. Given a planar embedding of a graph G = (V, E) with regions R, if |V| = 14 and |R| = 10, then find |E|. !13. Give an example of a planar graph G and two planar embeddings of G that give rise to non-isomorphic duals.
CHAPTER 9
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551
!42. Suppose that, from given a graph G, a graph G ! is obtained by combining two nonadjacent vertices of G into one. (a) Show: χ (G ! ) ≤ χ (G) + 1.
(b) Give an example for which χ (G ! ) < χ (G) + 1.
43. Prove Brooks’ Theorem in the case that G is a regular graph with κ(G) = 1. Hint: For each component H of G\{v}, optimally color the subgraph induced by H ∪ {v}. 44. Show that there always is some ordering of the vertices for which the Greedy Coloring Algorithm yields an optimal coloring. Hint: Let c = χ (G). Pick a coloring with c colors and color classes A1 , . . . , Ac such that |A1 | ≤ · · · ≤ |Ac | and A1 is smallest possible, A2 is smallest possible given A1 , A3 is smallest possible given A2 , and so on. Argue that for each i, each vertex of Ai is adjacent to every color j with j > i. Otherwise, Ai could be made smaller. Order the vertices so that Ac , . . . , A1 . 45. Pictured are the voting districts in a certain county.
The county commissioners want to include a color picture of this map in a pamphlet, so that colors clearly distinguish voting districts. Since each additional color adds expense to the production of this pamphlet, they want to minimize the number of colors needed. What is the smallest number of colors that can be used? 46. Pictured are the police precincts in a portion of a major city.
The mayor wants to create a color version of this map, with colors distinguishing precinct boundaries, using as few colors as possible. How many colors are required? 47. Find a map of the 48 contiguous states in the United States. (a) Find a coloring of the map using as few colors as possible so that no two adjacent states have the same color. (b) How many colors are needed? (c) Prove that no fewer colors would suffice. 48. Find a map of the 54 countries in Africa. (a) Find a coloring of the map using as few colors as possible so that no two adjacent countries have the same color. (b) How many colors are needed? (c) Prove that no fewer colors would suffice.
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Bradley Park
Comstock
Daniels
Main Winter
Market
Center
Routes 1 2
Hawk
Anselm
South Fern
Consider the graph representing this bus system. (a) Is there a unique path from Fern to Market? (b) Is there a unique path from Hawk to Park? (c) The graph is not a tree, so in general, paths in it are not unique. However, is there a unique shortest path between any pair of stops in this city? For this, we assume that all edges in our graph have the same length. Explain your answer. 14. Pictured are the roads in a particular neighborhood.
Consider the graph in which intersections (including dead ends) are represented by vertices, and streets joining them are represented by edges. (a) Observe that the graph is a tree. For any pair of intersections that sit in this neighborhood, must there be a unique path between them? (b) What features of the tree represent the dead ends? (c) What features of the tree represent the intersections? !15. Prove the converse of Theorem 10.1. !16. Prove the converse of Theorem 10.3 for connected graphs. !17. Let G be an arbitrary graph without loops. Show: G has only one spanning tree if and only if G is a tree. 18. Find all trees with degree sequence 4, 4, 2, 2, 1, 1. 19. Let G = (V, E) be an arbitrary graph. Show: If G is connected and |E| = |V| − 1, then G is a tree. Hint: Consider a spanning tree. !20. Use Theorem 10.3 to prove Theorem 10.2. Hint: Consider the degree sequence. !21. Show that a tree has exactly two leaves if and only if it is a path. 22. What is the degree sequence of any tree with exactly 3 leaves? 23. How many edges are there in a forest on n vertices with c components?
∗ Exercises
that may be particularly challenging are marked with a star.
CHAPTER 10
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Trees and Algorithms
573
For Exercises 47 through 50, let T be a given full m-ary tree. Let n denote the number of vertices, l the number of leaves, and i the number of internal vertices. Prove Theorem 10.6. 48. Express i and l in terms of n and m. Express n and i in terms of l and m. Express n and l in terms of i and m. Beth Robbins made a family tree consisting of the women in her family, starting with her grandmother. After doing so, she noticed that each mother has exactly three daughters and precisely ten of the women are mothers. How many women are in Bethâ&#x20AC;&#x2122;s family? 52. Luther Foote is trying to make a family tree consisting of the men in his family, starting with his father. If precisely eight of the men do not have sons but every father has exactly two sons, then how many men are in the Foote family?
!47. 49. 50. 51.
53. Let T be a full m-ary tree of height h in which all l leaves are at level h. Show that l = mh . !54. Let T be a full and balanced m-ary tree of height h with l leaves. Show that h = !logm l". In Exercises 55 through 58, construct the binary search tree for the words in the given sentence. Use the usual dictionary order to compare words. 55. All students take calculus. 56. Long live the king. 57. Eat, drink, and be merry. 58. Three is the magic number. In Exercises 59 through 62, construct the quadtree for the given image. Note that grid lines have been drawn over the image for reference and are not themselves part of the image. 59. 60.
61.
62.
In Exercises 63 through 66, draw the image that corresponds to the given quadtree. In each case, the image represents a letter. 63. W W B B W
W B B B
W W B B
64.
B B W B
B B B W
W B B B
B W B B
CHAPTER 10
9. 11. 13. 15. 17.
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Trees and Algorithms
Graph (b). 10. Graph (a). Graph (d). 12. Graph (c). Graph (f). 14. Graph (e). Graph (h). 16. Graph (g). Does a minimum spanning tree always contain an edge of smallest weight? Justify your answer.
18. Let G be a weighted graph with all weights distinct. Suppose C is a cycle in G. Is it possible that the largest-weight edge of C is in a minimum spanning tree for G? Justify your answer. !19. Suppose G is a weighted graph with all weights distinct. Prove or disprove that G must have a unique minimum spanning tree. 20. Let G be a weighted graph, and form a weighted graph G ! by either adding some fixed constant to all of the weights of G or multiplying all of the weights of G by some fixed positive constant. Show that, as a subgraph, a tree is a minimum spanning tree for G if and only if it is for G ! . 21. A college wishes to set up a closed circuit television network on its campus. The following table gives the costs determined by a contractor to connect pairs of buildings on the campus.
Bartle Evans Guild Miles Pine Taft Waller
Bartle
Evans
Guild
Miles
Pine
Taft
Waller
— $5000 $3000 — — — —
$5000 — $5000 $2000 — — —
$3000 $5000 — — $2000 $9000 —
— $2000 — — $4000 — $3500
— — $2000 $4000 — $6000 $4500
— — $9000 — $6000 — —
— — — $3500 $4500 — —
Find the cheapest cost of establishing a TV network that serves all the buildings. 22. A new summer camp needs to provide electricity to each of its buildings. The following table gives the costs of connecting pairs of buildings in this summer camp.
Assembly Cafeteria Counselors Dorm 1 Dorm 2 Dorm 3
Assembly
Cafeteria
Counselors
Dorm 1
Dorm 2
Dorm 3
— $4000 $5000 — — $5000
$4000 — $3000 $3000 — —
$5000 $3000 — — $4000 —
— $3000 — — $2000 $3000
— — $4000 $2000 — $2000
$5000 — — $3000 $2000 —
Find the cheapest cost of providing power to all the buildings in the camp.