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(a) Find a perfect matching in the graph below. 1 (b) If G = (V, E) has a perfect matching, show that [V] is even. (c) Find a perfect matching of Kg. How many perfect matchings are there of Kg?

(a) Find a perfect matching in the graph below. 1 (b) If G = (V, E) has a perfect matching, show that [V] is even. (c) Find a perfect matching of Kg. How many perfect matchings are there of Kg?

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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3. Let G = (V, E) be a simple undirected graph. A perfect matching of G is a matching M such that V is M-saturated, i.e. for each x V, there is an edge e M such that e contains x. For example, below on the left is K₁ and below on the right are the 3 perfect matchings of K4. AA A (a) Find a perfect matching in the graph below. 1 (b) If G = (V, E) has a perfect matching, show that IV is even. (c) Find a perfect matching of Kg. How many perfect matchings are there of Kg? (d) For n ≥ 1, let F(n) be the number of perfect matchings of K2n, the complete graph on 2n vertices. Find a recurrence relation for F(n).
Transcribed Image Text:3. Let G = (V, E) be a simple undirected graph. A perfect matching of G is a matching M such that V is M-saturated, i.e. for each x V, there is an edge e M such that e contains x. For example, below on the left is K₁ and below on the right are the 3 perfect matchings of K4. AA A (a) Find a perfect matching in the graph below. 1 (b) If G = (V, E) has a perfect matching, show that IV is even. (c) Find a perfect matching of Kg. How many perfect matchings are there of Kg? (d) For n ≥ 1, let F(n) be the number of perfect matchings of K2n, the complete graph on 2n vertices. Find a recurrence relation for F(n).
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