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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.

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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 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?

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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.
Transcribed Image Text: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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