Matchings in graphs
12.1 It is obvious that for a bipartite graph to contain a perfect matching, it is necessary that . Show that if every node has the same degree, then this is indeed so.
12.2 Show by examples that the conditions formulated in the theorem cannot be dropped:
(a) A non-bipartite graph in which every node has the same degree need not contain a perfect matching.
(b) A bipartite graph in which every node has positive degree (but not all the same) need not contain a perfect matching.
12.3 Prove Theorem 12.1 for and .
12.4 Prove that if in a bipartite graph every node has the same degree , then the bipartite graph is “good” (and hence contains a perfect matching; this proves theorem 12.1).
12.5 Show by an example that it may happen that a bipartite graph has a perfect matching but, if we are unlucky, the matching constructed above is not perfect.
12.6 Prove that if has a perfect matching, then matches up at least half of the nodes.
12.7 Follow how the algorithm works on the graph in Figure 38.
12.8 Show how the description of algorithm above contains a new proof of theorem 12.2.
12.9 Decide whether or not the graphs in Figure 39 have a Hamiltonian cycle.
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