Long paths we show that for every n ≥ 3 if deg(v) ≥ n/2 forevery v ∈ V then the graph contains a simple cycle (no vertexappears twice) that contains all vertices. Such a path is called anHamiltonian path. From now on we assume that deg(v) ≥ n/2 for everyv.
1. Show that the graph is connected (namely the distance betweenevery two vertices is finite)
2. Consider the longest simple path x0,x1, . . . , xk in the graph. Show that thereare at least n/2 vertices in {x1, x2, . . . ,xk} that are connected to x0, and at leastn/2
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