Expected Crossing Numbers [chapter]

Bojan Mohar, Tamon Stephen
2013 Fields Institute Communications  
Pach and Tóth [6] introduced a new version of the crossing number parameter, called the degenerate crossing number, by considering proper drawings of a graph in the plane and counting multiple crossing of edges through the same point as a single crossing when all pairwise crossings of edges at that point are transversal. We propose a related parameter, called the genus crossing number, where edges in the drawing need not be represented by simple arcs. This relaxation has two important
more » ... . First, the genus crossing number is invariant under taking subdivisions of edges and is also a minor-monotone graph invariant. Secondly, it is "computable" in many instances, which is a rare phenomenon in the theory of crossing numbers. These facts follow from the proof that the genus crossing number is indeed equal to the non-orientable genus of the graph. It remains an open question if the genus crossing number can be strictly smaller than the degenerate crossing number of Pach and Tóth. A relation to the minor crossing number introduced by Bokal, Fijavž, and Mohar [1] is also discussed.
doi:10.1007/978-3-319-00200-2_12 fatcat:fybo5nvjcnhg3ijlshhz5ultcu