Download
1 / 5
50 likes | 216 Vues
How many non-isomorphic tournaments with 10 vertices are there? a). 5 b). 10 c). 362,880 d). Over nine million. What is the first line of the proof? a). Assume d(v,u) 2 for all vertices u in a tournament T.
E N D
How many non-isomorphic tournaments with 10 vertices are there? a). 5 b). 10 c). 362,880 d). Over nine million
What is the first line of the proof? a). Assume d(v,u) 2 for all vertices u in a tournament T. b). Assume vertex v is such that d(v,u) 2 for all vertices u in a tournament T. c). Assume v is any vertex of maximum score in a tournament T. d). Let T be any tournament with a vertex of maximum score.
What is the next line of the proof? a). Assume the result holds for graphs with k vertices. b). Notice that (ui, uj) is an arc for some i and j. c). Notice that d(v,ui) = 1 for i = 1, 2, …, n. d). Notice that d(v,ui) = 2 for i = 1, 2, …, n. e). If p =1, note that the theorem holds.
What is the next line of the proof? a). Let u be any one of the ui’s. b). Let u be any vertex that is not adjacent from v. c). Notice that v has maximum score in T. d). Since d(v,ui) = 1 for all i, we have d(v,ui) 2 and we are done.
What is the next line of the proof? a). Then d(v,u) = 2. b). Then (v,u) is an arc in T. c). Then (u,v) is an arc in T. d). Assume d(v,u) > 2. e). Delete u from G to obtain G – u.
