4. Let n ∈ Z+ with n ≥ 4, and let the vertex set V ′ for the complete graph Kn−1 be{v1, v2, v3, . . . , vn−1}. Now construct the loop-free undirected graph Gn = (V, E) from Kn−1 asfollows: V = V ′ ∪ {v}, and E consists of all the edges in Kn−1 except for the edge {v1, v2}, whichis replaced by the pair of edges {v1, v} and {v, v2}.a) Determine deg(x) + deg(y) for all nonadjacent vertices x and y in V .b) Does Gn have a Hamilton cycle?c) How large is the edge set E?d) Do the results in parts (b) and (c) contradict Corollary 11.6?PS:Please do not use ChatGPT and type the correct answer! 4. Let nЄ Z+ with n ≥ 4, and let the vertex set V' for the complete graph K-1 be{V1, V2, V3, ..., Un−1}. Now construct the loop-free undirected graph G = (V,E) from KË−1 asfollows: V = V'U{v}, and E consists of all the edges in K-1 except for the edge {v1, v2}, whichis replaced by the pair of edges {v1, v} and {v, v2}.a) Determine deg(x) + deg(y) for all nonadjacent vertices x and y in V.b) Does G have a Hamilton cycle?c) How large is the edge set E?d) Do the results in parts (b) and (c) contradict Corollary 11.6?Corollary 11.6: If G = (V,E) is a loop-free undirected graph with |V| = n ≥ 3, and if|E| ≥ ("₂¹) +2, then G has a Hamilton cycle.

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Answered: 4. Let nЄ Z+ with n ≥ 4, and let the…

Fundamentals of Information Systems

8th Edition

ISBN:9781305082168

Author:Ralph Stair, George Reynolds

Publisher:Ralph Stair, George Reynolds

Chapter3: Database Systems And Applications

Section: Chapter Questions

Problem 2.1DQ

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Question

4. Let n ∈ Z+ with n ≥ 4, and let the vertex set V ′ for the complete graph Kn−1 be
{v₁, v₂, v₃, . . . , v_n−1}. Now construct the loop-free undirected graph G_n = (V, E) from K_n−1 as
follows: V = V ′ ∪ {v}, and E consists of all the edges in Kn−1 except for the edge {v₁, v₂}, which
is replaced by the pair of edges {v₁, v} and {v, v₂}.
a) Determine deg(x) + deg(y) for all nonadjacent vertices x and y in V .
b) Does Gn have a Hamilton cycle?
c) How large is the edge set E?
d) Do the results in parts (b) and (c) contradict Corollary 11.6?