Let G be a connected graph of order n and size n. Prove that G contains a single cycle.
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- Let G be a graph of order n ≥ 3. Prove that G is 2-connected if and only if for any vertex v and edge e of G, v and e lie on a common cycle of G.Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only if u and v are in the same connected component of G.Let G be a C7 graph (a circuit graph with 7 vertices). Prove that Gc has a Euler cycle.
- Suppose a graph G has the following Harmiltonian Cycle: BAGDFHEJKXB a. How many vertices does G have? b. What is the length of the cycle? C Find two distinct paths between D'and X (distinct means they use different edges) Type your answer in the space provided. For the toolbar, press ALT+F10 (PC) or ALT +FN+F10 (Mac).Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph obtained from G by creating a new edge, e, joining v and w. Prove that x(G) = min{x(G+e), x((G+e) 4e)}.