2. Find a counterexample to the converse of Theorem 11.8 and show the full answer. Theorem 11.8: Let G = (V, E) be a loop-free graph with |V | = n ≥ 2. If deg(x) +deg(y) ≥ n − 1 for all x, y ∈ V, x̸ = y, then G has a Hamilton path.
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2. Find a counterexample to the converse of Theorem 11.8 and show the full answer.
Theorem 11.8: Let G = (V, E) be a loop-free graph with |V | = n ≥ 2. If deg(x) +
deg(y) ≥ n − 1 for all x, y ∈ V, x̸ = y, then G has a Hamilton path.
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- 8. Answer these two questions:3.1.11. Let C and C' be cycles in a graph G. Prove that CaC' decomposes into cycles.Hey, The condensation of a graph G with k strong coherence components G1 =.(V1 , E1 ), . . . , Gk = (Vk , Ek )is the reduction of the original graphto its strong coherence components. In this case, the coherence components are combined into one node each in the condensation. The condensation to G is thus the graph G↓=({V1,...,Vk},E),where(Vi,Vj)∈E ⇔i̸=j∧∃u∈Vi,v∈Vj:(u,v)∈E holds. what is the Kondensation G↓ of the graph in the picture? Thank you in advance!
- please urgntly1. Let C(G, k) denote the decision problem of whether the undirected graph G = (V, E) has a subset of vertices V' C V such that |V'| = k and there is an edge connecting every pair of vertices in V'. Prove that C(G, k) is NP-Complete.4) a) Two bugs follow the paths ri(t) = and 20t-10 r2(t) =. Determine if their paths cross. If so, at what point? t+2
- (7)Recall that Kn,m is a complete bipartite graph. Prove that Kn,m is hamiltonian if and only if n = m and n > 2.(i) Let e1, e2, . . . , one connection of the graph G and let xi−1 and xi be the sites of the connection ei (1 ≤ i ≤ n). Show that every closed walk (e1, e2, . . . , en) is of length at least 3 with pairs of distinct pointsx1, x2, . . . , xn cycles.(ii) A graph containing no cycles is called acyclic. A walk is acyclic if the subgraph consisting of points and links of the walk is acyclic. Prove: a walk has all distinct points if and only if it is an acyclic sequence.(iii) If and are in different points of the graph G and if there is a walk in G from u to v, show that then there is an acyclic sequence from u to v.(Explain precisely that every shortest walk from u to v is actually an acyclic path.)Solve B
- 8.2-1ab)If X and Y are path-connected, thon So is X XYQuestion 4 Show that every u v walk in a graph contains au- v path. Question 5 [5.1] Prove or disprove that a graph and its complement cannot both be disconnected.. [5.2] Prove or disprove that if G is a connected graph, then its complement G is disconnected. Question 6 Consider the following graph G a b C d V}] e g h Determine the following (justify all your answers): 1. The order of G. 2. The size of G. 3. Two adjacent vertices in G. 4. Two nonadjacent vertices in G. 5. The open neighborhood of d. 6. The closed neighborhood of d. 7. The maximum degree of G. 8. The minimum degree of G. 9. The degree sequence of G. Question 7 During the Covid-19 lockdown, a group of 10 people met around a diner party. They each shook hands with each other, how many handshakes took place in that diner? Hint: Use vertices and edges. END