A-46 ANSWERS 23. a) OR WA ID MT WY b) Yes; WA ID MT WY ID , , , , , OR WA , 25. a) T N A S Q P J b) Yes; STNAPNQJSQT , , , , , , , , , , c) No 27. a) V M L T C b) Yes; T M L V C L T C , , , , , , , c) No 29. a) A B D F E C O b) Yes c) OABCDFCEOA , , , , , , , , , 31. a) A O C D B E b) No 33. a) Yes b) The residents would need to start at the intersection of Maple Cir., Walnut St., and Willow St. or at the intersection of Walnut St. and Oak St. 35. A D E C B A C D , , , , , , , 37. HIFCBDGHEDABEF , , , , , , , , , , , , , 39. ABCDIHGBEHCEGFA , , , , , , , , , , , , , , 41. ACDGHFCFEBA , , , , , , , , , , 43. ABCEBDEFIEHDGHIJFCA , , , , , , , , , , , , , , , , , , 45. UT CO NM AZ CA NV UT AZ NV , , , , , , , , 47. BAEHIJKDCGGJFCBFIEB , , , , , , , , , , , , , , , , , , 49. JGGCFJKDCBFIEBAEHIJ , , , , , , , , , , , , , , , , , , 51. a) Yes. There are no odd vertices. b) Yes. There are no odd vertices. 53. a) No. There are more than two odd vertices. b) No. There is at least one odd vertex. 55. a) No b) California, Nevada, and Louisiana (and others) have an odd number of states bordering them. Since a graph of the United States would have more than two odd vertices, no Euler path and no Euler circuit exist. 29. B A C D 31. OR WA ID MT WY 33. P E H N C B G 35. K O B F GR L D 37. GB GR FB K L B SB E DR G O 39. C A F E G B D 41. Answers will vary. 43. a) Answers will vary. b) Answers will vary. c) Answers will vary. d) The sum of the degrees is equal to twice the number of edges, which is true, since each edge must connect two vertices. Each edge then contributes two to the sum of the degrees. 44. B D K M Other answers are possible. L J 45. a) Answers will vary. b) Answers will vary. SECTION 13.2, PAGE 861 1. Euler 3. No 5. Odd 7. a) Yes. One example is A B D E C A D C , , , , , , , . b) No. This graph has exactly two odd vertices, A and C. Each Euler path must begin at vertex A and end at vertex C or vice versa. 9. No. A graph with exactly two odd vertices has no Euler circuits. 11. No. A graph with more than two odd vertices has neither an Euler path nor an Euler circuit. 13. No. A graph with more than two odd vertices has neither an Euler path nor an Euler circuit. 15. ABCDEFBDFA , , , , , , , , , 17. CDEFABDFBC , , , , , , , , , 19. EFABCDFBDE , , , , , , , , , 21. a) Yes b) They could start on either island and finish at the other.
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