MathDB

Suppose there are

n

points on the plane, no three of which are collinear. Draw

n-1

non-intersecting segments (except possibly at endpoints) between pairs of points, such that it is possible to travel between any two points by travelling along the segments. Such a configuration of points and segments is called a network. Given a network, we may assign labels from

1

n-1

to each segment such that each segment gets a different label. Define a spin as the following operation:

\bullet

Choose a point

v

and rotate the labels of its adjacent segments clockwise. Formally, let

e_1,e_2,\cdots,e_k

be the segments which contain

v

as an endpoint, sorted in clockwise order (it does not matter which segment we choose as

e_1

). Then, the label of

e_{i+1}

is replaced with the label of

e_{i}

simultaneously for all

1 \le i \le k

. (where

e_{k+1}=e_{1}

)
A network is nontrivial if there exists at least

2

points with at least

2

adjacent segments each. A network is versatile if any labeling of its segments can be obtained from any initial labeling using a finite amount of spins. Find all integers

n \ge 5

such that any nontrivial network with

n

points is versatile.
Proposed by Yeoh Zi Song

Problem Statement