Discrete Applied Math Seminar By Marcus Schaefer: The Degenerate Crossing Number and Higher-Genus Embeddings
Marcus Schaefer, DePaul University
The Degenerate Crossing Number and Higher-Genus Embeddings
Can we quantify how much topology a graph embedded in a surface needs to use? On the non-orientable surface N_k we can localize the topology into k crosscaps. We can then ask how often an edge has to pass through each of the k crosscaps.
Bojan Mohar conjectured, more than ten years ago, that every edge has to pass through each crosscap at most once. Mohar's conjecture is still open, but we can show that "at most twice" is sufficient.
Mohar originally phrased his conjecture as a problem on the degenerate crossing number, and we will motivate the problem from the crossing number point of view. The connection also gives us results for the orientable case and the bundled crossing numbers.
The talk will be accessible to any one with basic knowledge of undergraduate graph theory. Topological concepts will be introduced in the talk.
The Discrete Applied Math seminar will be held most Fridays from 3:30 p.m. - 4:30 p.m. on Zoom during the Spring 2022 semester. Contact the seminar organizer for joining details.
Discrete Applied Math seminar