CROSSING RELATIONSHIPS IN THE GENUS CARICA

The Genus Crossing Number

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 ed...

Genus, Treewidth, and Local Crossing Number

Planar Crossing Numbers of Genus g Graphs

Pach and Tóth [15] proved that any n-vertex graph of genus g and maximum degree d has a planar crossing number at most cdn, for a constant c > 1. We improve on this results by decreasing the bound to O(dgn), if g = o(n), and to O(g), otherwise, and also prove that our result is tight within a constant factor.

The Degenerate Crossing Number and Higher-Genus Embeddings

If a graph embeds in a surface with k crosscaps, does it always have an embedding in the same surface in which every edge passes through each crosscap at most once? This well-known open problem can be restated using crossing numbers: the degenerate crossing number, dcr(G), of G equals the smallest number k so that G has an embedding in a surface with k crosscaps in which every edge passes throu...

The Genus Botrychium and Its Relationships.