938 • CHAPTER 15 Graph Theory

15.1 EXERCISES

For Exercises 1-6, determine how many vertices and how many edges each graph has.

1. 2.

3. 4.

s.

7.-10. For Exercises 7-10, refer to the graphs shown in Exercises 1-4. For each of these graphs. find the degree of each vertex in the graph. Then add the degrees to get the sum of the degrees of the ver­tices of the graph. What relationship do you notice between the sum of degrees and the num­ber of edges?

In Exercises 11-16, determine whether the two graphs are isomorphic. If so, label corresponding vertices of the two graphs with the same letters and color-code corresponding edges, as in Example 2. (Note that there is more than one correct answer for many ofthese exercises.)

11.

U (a) (b)

12.

lLJ (a) (b)

13.

14.

(a) (b)

15.

16.

(al

(a)

(b)

. (b)

L

(a) (b)

In Exercises 17-22, determine whether the graph is con­nected or disconnected. Then determine how many com­ponents the graph has.

18.£.7

19.

15.1 Basic Concepts • 939

20.

21. • 22.

In Exercises 23-26, use the theorem that relates the sum of degrees to the number of edges to determine the num­ber ofedges in the graph (without drawing the graph).

23. A graph with 5 vertices, each of degree 4

24. A graph with 7 vertices, each of degree 4

25. A graph with 5 vertices, three of degree I, one of degree 2, and one of degree 3

26. A graph with 8 vertices, two of degree I, three of degree 2, one of degree 3, one of degree 5, and one of degree 6

In Exercises 27-29, refer to the following graph.

A

F B~--",-""'G

c E

o

27. Which are walks in the graph? If not, why not? (a) A-B-C (b) B-A-D (c) E-F-A-E (d) B-D-F-B-D (e) D-E (f) C-B-C-B

28. Which are paths in the graph? Ifnot, why not? (a) B-D-E-F (b) D-F-B-D (c) B-D-F-B-D (d) D-E-F-G-F-D (e) B-C-D-B-A (f) A-B-E-F-A

29. Which are circuits in the graph? If not, why not? (a) A-B-C-D-E-F (b) A-B-D-E-F-A (c) C-F-E-D-C (d) G-F-D-E-F (e) F-D-F-E-D-F

In Exercises 30 and 31, refer to the following graph.

B J

c

30. Which are walks in the graph? If not, why not? (a) F-G-J-H-F (b) D- F (c) B-A-D-F-H (d) B-A-D-E-D-F-H (e) 1-G-J (f) 1-G-J-I

31. Which are paths in the graph? Ifnot, why not? (a) A-B-C (b) J-G-I-G-F (c) D-E~I-G-F

(d) C- A (e) C-A-D-E (f) C-A-D-E-D-A-B

In Exercises 32-37, refer to the following graph. In each case, determine whether the sequence of vertices is (i) a walk, (ii) a path. (iii) a circuit in the graph.

A

B E

c

32. A-B-C-D-E

33. A-B-C

34. A-B-C-D-A

35. A-B-A-C-D-A

~.A-B-C-A-D-C-E-A

37. C-A-B-C-D-A-E

In Exercises 59 through 64 on the next page, color the graph using as few colors as possible. Determine the chromatic number of the graph. (Hint: the chromatic num­ber is fixed, but there may be more than one correct coloring.)

942 0 CHAPTER 15 Graph Theory

6'. ~ 62...._+_~

63.

65. Color each graph using as few colors as possible. Use this to determine the chromatic number of the graph. (Graphs like these are called cycles.)

(e) Use your results from parts (a)-(d) to make a pre­diction about the chromatic number of a cycle. (Hint: consider two cases.)

66. Sketch a complete graph with the specified number of vertices, color the graph with as few colors as possi­ble, and use this to determine the chromatic number of the graph. (a) 3 vertices (b) 4 vertices (c) 5 vertices (d) Write a general principle by completing this

statement: A complete graph with n vertices has chromatic number

(e) Why is the statement in part (d) true? (1) Generalize further by completing this statement:

If a graph has a subgraph that is a complete graph with n vertices, then the chromatic number of the graph must be at least __'

In Exercises 67-70, color the vertices using as few colors as possible. Then state the chromatic number of the graph. (Hint: it might help to first identify the largest sub­graph that is a complete graph.)67.& 6•• 0) 6'.~

70.

71. Scheduling Meeting Times At a college, Campus Life must schedule weekly meeting times for the six organizations listed below in such a way that organi­zations with members in common meet at different times. Use graph coloring to determine the least num­ber of different meeting times and to decide which organizations should meet at the same time.

Organization Members also belong to

Choir Caribbean Club. Dance Club, Theater, Service Club

Caribbean Club Dance Club, Service Club, Choir

Service Club Forensics, Choir, Theater, Caribbean Club

Forensics Dance Club, Service Club Theater Choir, Service Club Dance Club Choir, Caribbean Club,

Forensics

72. Assigning Frequencies to Transmitters Interference can occur between radio stations. To avoid this, trans­mitters that are less than 60 miles apart must be assigned different broadcast frequencies. The trans­mitters for 7 radio stations are labeled A through G.

15.2 EXERCISES

In Exercises 1-3, a graph is shown and some sequences of vertices are specified. Determine which ofthese sequences show Euler circuits. Ifnot, explain why not.

1. A D

B C

OOA-B-C-D-A-B-C-D-A (b) C-B-A-D-C (c) A-C-D-B-A (d) A-B-C-D

2. A

B F

C E

D

(a) A-B-C-D-E-F-A (b) F-B-D-B (c) A-B-C-D-E­

F-B-D-F-A (d) A-B-F-D-B­

C-D-E-F-A

3. F E

A G D

B C

(a) A-B-C-D-E-F-A OOA-B-C-D-E-G-C-E­

F-G-B-F-A ~A-B-C-D-E-C-G-E­

F-G-E-F-A OOA-B-G-E-D-C-G-F­

B-C-E-F-A

In Exercises 4-8, use Euler's theorem to decide whether the graph has an- Euler circuit. (Do not actually find an Euler circuit.) Justify each answer briefly.

4. A D s. E

A F E

B C

B c D G

In Exercises 9 and /0, use Euler's theorem to determine whether it is possible to begin and end at the same place, trace the pattern without lifting your pencil, and trace over no line in the pattern more than once.

9.

10.

In Exercises //-15, use Euler's theorem to determine whether the graph has an Euler circuit, justifying each answer. Then determine whether the graph has a circuit that visits each vertex exactly once, except that it returns

15.

c

In Exercises 16-19 different floor tilings are shown. The material applied between tiles is called grout. For which ofthese floor tilings could the grout be applied beginning and ending at the same place, without going over any section twice, and without lifting the tool? Justify answers.

16. ~-~---r--C-----:I

968 • CHAPTER 15 Graph Theory

15.3 EXERCISES

In Exercises J and 2, a graph is shown. and some paths in OOA~B~C~D~E~F~G~A

the graph are specified. Determine which paths are OOB~I~G~F~E~D~H~F~

Hamilton circuits/or the graph. Ifnot. say why not. I~D~C~B~G~A~B

(c) A~B~C~D~E~F~G~H~1. A

I~A

B E In Exercises 5-10, determine whether the graph has a Hamilton circuit. If so. find one. (There are many different C D correct answers.) (a) A~E~C~D~E~B~A

(b) A~E~C~D~B~A

(c) D~B~E~A~B 5.~ (d) E~D~C~B~E

2. F

B D

AqF.----~E

6. G

A B ~F---.D

C

OOA~B~C~D~E~C~A~E~ D F1>cF~A

(b) A~C~ D~ E~F~ A B

(c) F~A~C~E~F E (d) C~D~E~F~A~B

In Exercises 3 and 4, determine whether the string 0/ 7. G vertices is a circuit, whether it is an Euler circuit, and whether it is a Hamilton circuit. Justify your answers.

3. E ~I A__--+~~-..."D H

8. A F E

B C

(a) A~B~C~D~E~A

(b) B~E~C~D~A~B

(c) E~B~A~D~A~D~C~E B C D

4. A

9. x

wT IF----~iIl F

C

D E u v

10. A E

B c D

11. Draw a graph that has a Hamilton circuit, but no Euler circuit. Specify the Hamilton circuit, and explain why the graph has no Euler circuit. (There are many differ­ent correct answers.)

12. Draw a graph that has an Euler circuit, but no Hamilton circuit. Specify an Euler circuit in your graph. (There are many different correct answers.)

13. Draw a graph that has both an Euler circuit and a Hamilton circuit. Specify these circuits. (There are many different correct answers.)

14. Decide whether each statement is true or false. If the statement is false, give an example to show that it is false. (a) A Hamilton circuit for a graph must visit each

vertex in the graph. (b) An Euler circuit for a graph must visit each ver­

tex in the graph. (c) A Hamilton circuit for a graph must use each

edge in the graph. (d) An Euler circuit for a graph must use each edge

in the graph. (e) A circuit cannot be both a Hamilton circuit and

an Euler circuit. (0 An Euler circuit must visit no vertex more than

once, except the vertex where the circuit begins and ends.

In Exercises 15-20, determine whether an Euler circuit, a Hamilton circuit, or neither would solve the problem.

15. Bandstands at a Festival The vertices of a graph represent bandstands at a festival and the edges rep­resent paths between the bandstands. A visitor wants to visit each bandstand exactly once, return­ing to her starting point when she is finished.

16. Relay Team Running Order The vertices of a com­plete graph represent the members of a five-person relay team. The team manager wants a circuit that will show the order in which the team members will run. (He will decide later who will start.)

17. Paths in a Botanical Garden The vertices of a graph represent places where paths in a botanical garden cross and the edges represent the paths. A visitor

15.3 Hamilton Circuits and Algorithms • 969

wants to walk along each path in the garden exactly once, returning to his starting point when finished.

18. Traveling in Western Europe The vertices of a graph represent the countries on the continent (Western Europe), with edges representing border crossings between the countries. A traveler wants to travel over each border crossing exactly once, returning to the first country visited for his flight home to the United States.

19. Traveling in Africa Vertices represent countries in sub-Saharan Africa, with an edge between two ver­tices if those countries have a common border. A trav­eler wants to visit each country exactly once, returning to the first country visited for her flight home to the United States.

20. Reading X-Rays In using X-rays to analyze the .structure of crystals, an X-ray diffractometer ­measures the intensity of reflected radiation from the crystal in thousands of different positions. Consider the complete graph with vertices repre­senting the positions where measurements must be taken. The researcher must decide the order in which to take these readings, with the diffractome­ter returning to its starting point when finished.

In Exercises 21-24, use your calculator; if necessary, to find the value.

21.4! 22.6!

23.9! 24. 14!

In Exercises 25-28, determine how many Hamilton cir­cuits there are in a complete graph with this number of vertices. (Leave answers infactorial notation.)

25. 10 vertices 26. 15 vertices

27. 18 vertices 28. 60 vertices

29. List all Hamilton circuits in the graph which start at P.

.-----.s

R

15.4 Trees and Minimum Spanning Trees • 981

I.EXAMPLE 5 Finding a Graph with Specified Properties---- _.•._'--­Suppose a graph is a tree and has 15 vertices. What is the greatest number of vertices of degree 5 that this graph could have? Draw such a tree.

SOLUTION

Because the graph is a tree with 15 vertices, it must have 14 edges. Do we know any­thing about graphs that will help us relate the number of edges to the degrees of ver­tices? Recall that the sum of the degrees of the vertices of a graph is twice the number of edges. Thus, the sum of the degrees of the vertices in the tree must be 2 • 14, which is 28.

Now we use trial and error. We want the greatest possible number of vertices with degree 5. Four vertices with degree 5 would contribute 20 to the total degree sum. This would leave 28 - 20 = 8 as the degree sum of the remaining II vertices, but this would mean that some of those vertices would have no edges joined to them. The graph would not be connected and would not be'a tree.

Let's try 3 vertices with degree 5. That would contribute 15 to the total degree sum; this would leave 28 - 15 = 13 as the degree sum for the remaining 12 vertices. After a little thought we see that we can draw a graph with these specifications. See Figure 54. Thus, the greatest possible number of vertices of degree 5 in a tree with 15 vertices is 3.FIGURE 54

15.4 EXERCISES

In Exercises 1-7, determine whether the graph is a tree. If 6. a complete graph with 6 vertices not, explain why it is not.

7. a connected graph with all vertices having even degree

0

2 ,? In Exercises 8-10, add edges (no vertices) to the graph to change the graph into a tree, or explain why this is not pos­sible. (Note: There may be more than one correct answer.)

8. A B C 9. K

E D M

G H L F P

N 0

1.

10. Q RS0l@ TS·

u

982 • CHAPTER 15 Graph Theory

In Exercises 11-13, determine whether the graph described must be a tree.

11. Spread of a Rumor A sociologist is investigating the spread of a rumor. He finds that 18 people know of the rumor. One of these people must have started the rumor, and each must have heard the rumor for the first time from one of the others. He draws a graph with vertices representing the 18 people and edges showing from whom each person first heard the rumor.

12. Interlfet Seorch Engine You are using the Google search engine to search the Web for information on a topic for a paper. You start at the Google page and fol­low links you find, sometimes going back to a site you've already visited. To keep a record of the sites you visit, you show each site as a vertex of a graph; you draw an edge each time you connect for the first time to a new sileo

13. Trocking Infectious Oiseose A patient has a highly infectious disease. An employee of the Centers for Disease Control is trying to quarantine all people who have had contact over the past week with either the patient or with someone already included in the con- . tact network. The employee draws a graph with ver­tices representing the patient and all people who have had contact as described. The edges of the graph rep­resent the relation "the two people have had contact."

In Exercises 14-17, determine whether the statement is true or false. If the statement is false, draw an example to show that it is false.

14. Every graph with no circuits is a tree.

15. Every connected graph in which each edge is a cut edge is a tree.

16. Every graph in which there is a path between each pair of vertices is a tree.

17. Every graph in which each edge is a cut edge is a tree.

In Exercises 18-20, find three different spanning treesfor each graph. (There are many different correct answers.)

19'A~

B~D C

20. B C

A<IX1>D F E

In Exercises 21-23, find all spanning trees for the graph.

21. A D F

I U---..B C E G

22.A~G

vc D~ B F

23. L'S/D B C

In Exercises 24-26, determine how many spanning trees the graph has.

24'LLL\ 25.~

26.

27. What is a general principle about the number of spanning trees in graphs such as those in Exer­cises 24-26?

15.4 Trees and Minimum Spanning Trees • 983

28. Complete the parts of this exercise in order. 30. (a) Find all the spanning trees of each of the following

graphs.

(i) (ii) 14

31. D

Afk---tr

28

(iii)

(b) For each of the following graphs, determine how many spanning trees the graph has.

32. A

iii

c 9 D

(ii)

33. School Building Layout A school consists of 6 sepa­(iii) rate buildings, represented by the vertices in the fol­

lowing graph. (c) What is a general principle about the number of

spanning trees in graphs of the kind shown in (a) 32 and (b)?

In Exercises 29-32, use Kruskal's algorithm to find a minimum spanning tree for the graph. Find the total weight of this minimum spanning tree.

29. G 2 F

DA E There are paths between some of the buildings as

10 shown. The graph also shows the length in feet of each path. School administrators want to cover some

B 20 C of these paths with roofs so that students will be able