MathDB

There are

n

points in the plane, some of which are connected by segments. Some of the segments are colored in white, while the others are colored black in such a way that there exist a completely white as well as a completely black closed broken line of segments, each of them passing through every one of the

n

points exactly once. It is known that the segments

AB

and

BC

are white. Prove that it is possible to recolor the segments in red and blue in such a way that

AB

and

BC

are recolored as red, [hide=not all of which segments are recolored red]meaning that recoloring every white as red and every black as blue is not acceptable, and that there exist a completely red as well as a completely blue closed broken line of segments, each of them passing through every one of the

n

points exactly once.

Problem Statement