Hello, everyone!
In Projective Plane Crossing Number -- from Wolfram MathWorld claims that “All graphs with graph crossing number 0 or 1 (i.e., planar and singlecross graphs) have projective plane crossing number 0,” but I don’t think this is true.
No matter how hard I try to prove or disprove this statement on a Möbius strip with a “bottom,” I can’t. Perhaps you know how any graph with planar crossing number 1 can be laid out on the projective plane without crossings?
I think the statement is correct, but the key point may be that a single crossing in the plane can be removed by using the extra topology of the projective plane.
A graph with crossing number 1 has a drawing where only two edges cross. On the projective plane, that crossing can potentially be replaced by routing one of the edges through the crosscap, giving a crossing-free embedding.
I was initially thinking about the Möbius-strip model too, and I think the important part is how the boundary is identified when turning the Möbius strip into the projective plane. It would be interesting to see an explicit construction for an arbitrary singlecross graph, rather than just a particular example.
For example, the Petersen graph on the projective plane.
I might be wrong, but as far as I understand the structure of the projective plane, by routing one of the edges through the crosscap (passing through the equator?) we end up on the inside (crosscap is not a handle). I know the surface is one-sided, but the idea is this: hold the Möbius strip between your thumb and index finger, if your thumb touches the point where we ended up, then your index finger will touch the point where we should have ended up.
I can’t understand this example. How to paint K6 on the Cross-Cap?
Hey guys, I recently discovered (correct me if I’m wrong) that a projective plane is a sphere with one hole! One! This means that when drawing a graph, if we want to use the interior of the sphere, we’ll have to dive into this hole, and to emerge again, we’ll have to somehow intersect the previously drawn edge at the point where we dive. Therefore, the statement in the opening post may be wrong, but it clearly holds for a sphere with two holes, that is, a Klein bottle. How can I contact the author of the quote from the first post to find out their point of view?