Question

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

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

4. Show that the the graph contains an Hamiltonian cycle

Please give time complexity.

Answer 1