The genus crossing number

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

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...

Analogies between the crossing number and the tangle crossing number

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tangl...

Odd Crossing Number Is Not Crossing Number

The crossing number of a graph is the minimum number of edge intersections in a plane drawing of a graph, where each intersection is counted separately. If instead we count the number of pairs of edges that intersect an odd number of times, we obtain the odd crossing number. We show that there is a graph for which these two concepts differ, answering a well-known open question on crossing numbe...