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2. Let G be a connected graph. For all vertices u and v in G, we define the distance d(u, v) to be the least number d such that there is a path between u and v of length d. Prove that for all vertices u, v and w, d(u, w) ? d(u, v) + d(v, w). 3. Let G be a bipartite graph. Prove that if v1, . . . , vn is a cycle in G, then n is odd (so the cycle has even length). 4. (a) Let T be a tree. Prove that if you remove a leaf l and the only edge incident with l from T, the resulting subgraph T' is still a tree. (b) Prove by induction on n ? 1 that if a tree T has n vertices, then it has exactly n ? 1 edges. (c) Prove that if G is a connected graph with n vertices, then it has at least n ? 1 edges. (Hint: Spanning tree.)

          2. Let G be a connected graph. For all vertices u and v in G, we define the distance d(u, v) to be the least number d such that there is a path between u and v of length d. Prove that for all vertices u, v and w,
d(u, w) ? d(u, v) + d(v, w).
3. Let G be a bipartite graph. Prove that if v1, . . . , vn is a cycle in G, then n is odd (so the cycle has even length).
4. (a) Let T be a tree. Prove that if you remove a leaf l and the only edge incident with l from T, the resulting subgraph T' is still a tree.
(b) Prove by induction on n ? 1 that if a tree T has n vertices, then it has exactly n ? 1 edges.
(c) Prove that if G is a connected graph with n vertices, then it has at least n ? 1 edges. (Hint: Spanning tree.)

2. Let G be a connected graph. For all vertices u and v in G, we define the distance d(u, v) to be the least number d such that there is a path between u and v of length d. Prove that for all vertices u, v and w,
d(u, w) ? d(u, v) + d(v, w).
3. Let G be a bipartite graph. Prove that if v1, . . . , vn is a cycle in G, then n is odd (so the cycle has even length).
4. (a) Let T be a tree. Prove that if you remove a leaf l and the only edge incident with l from T, the resulting subgraph T' is still a tree.
(b) Prove by induction on n ? 1 that if a tree T has n vertices, then it has exactly n ? 1 edges.
(c) Prove that if G is a connected graph with n vertices, then it has at least n ? 1 edges. (Hint: Spanning tree.)

Added by Juan Manuel M.

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