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Finding a Star of Equal Degrees

Source: Iranian Combinatorics Olympiad P4

April 25, 2020
combinatoricsgraph theorygraph

Problem Statement

Given a graph with 9999 vertices and degrees in {81,82,,90}\{81,82,\dots,90\}, prove that there exist 1010 vertices of this graph with equal degrees and a common neighbour.
Proposed by Alireza Alipour