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Proof Techniques

Proof Techniques. Induction. If u and v are distinct vertices in G, then every u,v-walk in G contains a u,v-path. Every closed odd walk contains an odd cycle. Contra-positive. We use: (B  A)  (A  B)

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Proof Techniques

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  1. Proof Techniques

  2. Induction • If u and v are distinct vertices in G, then every u,v-walk in G contains a u,v-path. • Every closed odd walk contains an odd cycle.

  3. Contra-positive We use: (B  A)  (A  B) • A graph is connected iff for every partition of its vertices into two non-empty sets, there is an edge with endpoints in both sets • An edge of a graph is a cut-edge iff it belongs to no cycle

  4. Contradiction We prove A  B by showing that “A true and B false” is impossible • Suppose G has a vertex set {v1, …, vn}, with n3. If at least two of the subgraphs from Gv1, …, Gvn are connected, then G is connected • A graph is bipartite iff it has no odd cycle

  5. Extremality • If G is a simple graph in which every vertex degree is at least k, then G contains a path of length at least k. • If k2, then G also contains a cycle of length at least k+1. • If G is a nontrivial graph and has no cycle, then G has a vertex of degree 1. • Every nontrivial graph has at least two vertices that are not cut vertices.

  6. TheReconstruction Conjecture • A graph G is reconstructible if we can reconstruct G from the list { Gvi: vi  V(G) } • The famous unsolved reconstruction conjecture says that every graph with at least three vertices is reconstructible.

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