Exercises – 2 points
1. Show that every automorphism of a tree fixes a vertex or an edge.
2. Show that the cycle space of a graph is generated by its geodesic cycles.
3. In a graph G on n vertices and k connected components, show that there are n k vertices v1, . . . , vn−k such that the cuts E(vi) are linearly independent in the cut space of G. Use this to prove that the cut space is the orthogonal complement of the cycle space.
4. Let M be a matching on non-maximum size in a bipartite graph. Show that there exists an augmenting path with respect to M . Use this to describe an efficient algorithm for finding a maximum matching in bipartite graphs.
5. Show that all stable matchings of a preference-ordered graph have the same size.
Problems – 5 points
1. Let G have a path P of length k, and a cycle that conta√ins both end-
points of P . Show that G has a cycle of length at least k.
2. Show that, if there are injections A → B and B → A for two sets A and B, then there is a bijection A ↔ B.1 Hint: Consider two bipartite graphs on A H B.
1Recall that the set theoretic definitions of cardinality say that |A| ≤ |B| if there is an injection A → B, and |A| = |B| if there is a bijection A ↔ B. Problem 2 proves that these definitions do not violently contradict the normal meaning of the symbol ≤.
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