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Assignment 10
MATH 239 Due: Wednesday, April 14 at 4:00pm Waterloo time A-10-1. {10 marks} Let G be a graph with a perfect matching, and such that all vertices in G have degree at least 2. (a) Prove that if G is bipartite, then G has at least two perfect matchings. (b) Show that (a) is not true if we do not require G to be bipartite. A-10-2. {10 marks} Let G be a bipartite graph with bipartition (A,B) with deg(u) ≥ 1 for all u ∈ A, and with deg(u) ≥ deg(v) for every edge uv with u ∈ A. Prove that G has a matching that saturates A. A-10-3. {10 marks} Give examples of the following: (a) A graph G with no perfect matching, such that every vertex of G has degree exactly 3. (b) A graph with exactly three perfect matchings. (c) A graph G with a perfect matching such that every vertex has degree at least 3 and G has no bridge, but also, G has no Hamilton cycle. For each part, you should prove that your graph satisfies all of the conditions.