Problems

In the picture below you can see the graphs of \(K_5\), the complete graph on \(5\) vertices, and \(K_{3,3}\), the complete bipartite graph on \(3\) and \(3\) vertices. A theorem states that these graphs cannot be embedded into plane, namely one cannot draw graphs \(K_5\) and \(K_{3,3}\) on a plane in such a way that there are no intersecting edges.
The question is: can you draw the graphs \(K_5\) and \(K_{3,3}\) without intersecting edges on a torus?

In chess, knights can move one square in one direction and two squares in a perpendicular direction. This is often seen as an ‘L’ shape on a regular chessboard. A closed knight’s tour is a path where the knight visits every square on the board exactly once, and can get to the first square from the last square.

This is a closed knight’s tour on a \(6\times6\) chessboard.

Can you draw a closed knight’s tour on a \(3\times3\) torus? That is, a \(3\times3\) square with both pairs of opposite sides identified in the same direction, like the diagram below.

Is it possible to link three rings together in such a way that they cannot be separate from each other, but if you remove any ring, then the other two will fall apart?