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Source: Ukrainian TST 2008 Problem 7

February 12, 2009
inductionpigeonhole principlenumber theorycombinatorics unsolvedcombinatorics

Problem Statement

There is graph G0 G_0 on vertices A1,A2,…,An A_1, A_2, \ldots, A_n. Graph G_{n \plus{} 1} on vertices A1,A2,…,An A_1, A_2, \ldots, A_n is constructed by the rule: Ai A_i and Aj A_j are joined only if in graph Gn G_n there is a vertices Ak≠Ai,Aj A_k\neq A_i, A_j such that Ak A_k is joined with both Ai A_i and Aj A_j. Prove that the sequence {Gn}n∈N \{G_n\}_{n\in\mathbb{N}} is periodic after some term with period T≤2n T \le 2^n.
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