Let G be a simple graph with n vertices, and suppose that d(v) + d(w) ? n - 1 for all pairs of distinct non-adjacent vertices v, w. Show that any two vertices of G are connected by a path of length at most 2. (In particular, G is connected.)
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But then their degrees would add up to less than n-1, contradicting the given condition. Now, let's choose two arbitrary vertices u and v in G. If they are adjacent, then there is a path of length 1 between them. Otherwise, they are non-adjacent, and we can apply Show more…
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