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Exercise11.17.a) Howmany paths of length 5 are there in the

Exercise11.17.a) Howmany paths of length 5 are there in the

Exercise11.17.a) Howmany paths of length 5 are there in the completebipartite graph K3,7? (Rememberthat a path suchas v1 →v2 →v3 →v4 →v5 →v6 is considered to be thesame as the path v6 →v5 →v4 →v3 →v2 →v1.)How many paths of length p are there in the complete bipartitegraph Km,n?8.LetX _ {1, 2, 3, . . . , n}, where n ≥ 2. Construct theloopfreeundirected graph G _ (V , E) asfollows:• (V ): Each two-element subset of X determines a vertexof G.• (E): If v1, v2 ∈V correspond to subsets {a, b} and {c, d},respectively, of X, draw the edge {v1,v2} in G when{a, b} ∩ {c, d} _ ∅.a)Showthat G is an isolated vertexwhen n _ 2 and thatGisdisconnected for n _ 3, 4.b)Showthat for n ≥ 5, G isconnected. (In fact, for allv1, v2 ∈V , either {v1, v2} ∈E or there is apath of length 2connecting v1 and v2.)6.Findall (loop-free) nonisomorphic undirected graphs withfour vertices. How many of these graphsare connected?Exercise 11.213.LetG be a cycle on n vertices. Prove that G is selfcomplementaryif and only if n _ 5.Exercise 11.321.Determinethe value(s) of n for which thecomplete graphKnhasan Euler circuit. For which n doesKn have an Euler trailbut not an Euler circuit?Exercise 11.414.Determinewhich of the graphs in Fig. 11.69 are planar. Ifa graph is planar, redraw it with noedges overlapping. If it isnonplanar, find a subgraph homeomorphicto either K5 or K3,3.Answer:Exercise 11.57.a) Forn ≥ 3, how many different Hamilton cycles are therein the complete graph Kn?b)Howmany edge-disjoint Hamilton cycles are there inK21c)Nineteenstudents in a nursery school play a game eachday where they hold hands to form acircle. For how manydays can they do this with no studentholding hands withthe same playmate twice?Exercise 12.17.Givean example of an undirected graphG _(V , E) where|V | _ |E| + 1 but G isnot a tree.

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