Closed path and parallel edges
Source: Swiss Math Olympiad 2010 - final round, problem 5
March 16, 2010
combinatorics proposedcombinatorics
Problem Statement
Some sides and diagonals of a regular -gon form a connected path that visits each vertex exactly once. A parallel pair of edges is a pair of two different parallel edges of the path. Prove that
(a) if is even, there is at least one parallel pair.
(b) if is odd, there can't be one single parallel pair.