Pigeonhole Principle and the Probabilistic Method
نویسنده
چکیده
As a story, this means that at a party with n persons, there exist two persons who know the same number of people at the party. Proof. For any graph on n vertices, the degrees are integers between 0 and n − 1. Therefore, the only way all degrees could be different is that there is exactly one vertex of each possible degree. In particular, there is a vertex v of degree 0 (with no neighbors) and a vertex w of degree n − 1 adjacent to all other vertices). However, if there is an edge (v, w), then v cannot have degree 0, and if there is no edge (v, w) then w cannot have degree n− 1. This is a contradiction.
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