Problem 4 Let G = (V, E) be an undirected graph (and while it should go without saying, G has no self-loops or parallel edges). As always, let n denote the number of vertices in the graph, and m denote the number of edges; that is, n = |V| and m = |E|. (a) Use the extended Pigeonhole principle to prove that there is some vertex v € V with degree(v) > [2m]. (b) Use part (a) to show that m < n(n-1) 2

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