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Long paths we show that for every n ≥ 3 if deg(v) ≥ n/2 for every...

Long paths we show that for every n ≥ 3 if deg(v) ≥ n/2 for every v ∈ V then the graph contains a simple cycle (no vertex appears twice) that contains all vertices. Such a path is called an Hamiltonian path. From now on we assume that deg(v) ≥ n/2 for every v.

1. Show that the graph is connected (namely the distance between every two vertices is finite)

2. Consider the longest simple path x0, x1, . . . , xk in the graph. Show that there are at least n/2 vertices in {x1, x2, . . . , xk} that are connected to x0, and at least n/2 vertices of {x0, x1, . . . , xk−1} are neighbors of xk.

3. Show that there is an i so that x0 → xi+1 ∈ E and xi → xk both belong to E.

4. Show that the the graph contains an Hamiltonian cycle

Please give time complexity.

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