 # 1. Let G be a bipartite graph with partite sets A and B. A matching...

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1. Let G be a bipartite graph with partite sets A and B. A matching in G is a collection of edges sharing no endpoints in common, in other words, a matching is the set of edges of a subgraph in which every vertex has degree 1. Let R and B be two matchings in G and let H be the subgraph of G consisting of the edges R UB. Answer the following questions with a word or two or maybe a sentence, no proofs are necessary. (a) If A > B| , then what is the largest possible size of a matching in G? (b) What are the possible degrees of the vertices in H? (c) A connected component of H must be one of two types of graph. What are these two types? (d) What is special about the lengths of the cycles in H? (e) Must there be at least one connected components of H that is a path? (f) If R > B , then must there be at least one connected components of H that is a path? (g) If R > B , then must there be at least one connected components of H that is a path of odd length? (h) If R > B , then must there be at least one connected components of H that is a path of odd length whose first and last edges are both from R? (This includes the possibility of a path of length 1.)

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