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 as follows: V = V'U{v}, and E consists of all the edges in K-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 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.

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?

PS:Please do not use ChatGPT and type the correct answer!