Let G = (V0; V1; E) be a connected bipartite graph, where V0; V1 is the bipartition of the vertex set V (G) into independent sets. A bipartite drawing of G consists of placing the vertices of V0 and V1 into distinct points on two parallel lines x0 ; x1, respectively, and then drawing each edge with one straight line segment which connects the points of x0 and x1 where the endvertices of the edg...

The crossing number cr(G) of a graphG is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [RT93] conjectured that there is a constant c such that every graph G with crossing number k has an edge e such that cr(G− e) ≥ k− c √ k. They showed only that G always has an edge e with cr(G − e) ≥ 2 5 cr(G) − O(1). We prove that for every fixed ǫ > 0, t...

It is proved that the rectilinear crossing number of every graph with bounded tree-width and bounded degree is linear in the num-

We generalise theorems of Cochran-Lickorish and Owens-Strle to the case of links with more than one component. This enables the use of linking forms on double branched covers, Heegaard Floer correction terms, and Donaldson’s diagonalisation theorem to complete the table of unlinking numbers for nonsplit prime links with crossing number nine or less.

A book with k pages consists of a straight line (the spine) and k half-planes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The k-page crossing number νk(G) of a graph G is the minimu...

The crossing number of a graph is the least number of crossings of edges among all drawings of the graph in the plane. In this article, we prove that the crossing number of the generalized Petersen graph P (10, 3) is equal to 6.

In this paper we consider faultfree tromino tilings of rectangles and characterize rectangles that admit such tilings. We introduce the notion of crossing numbers for tilings and derive bounds on the crossing numbers of faultfree tilings. We develop an iterative scheme for generating faultfree tromino tilings for rectangles and derive the closed form expression for the exact number of faultfree...

In various research papers, such as [2], one will find the claim that the minLA is optimally solvable on outerplanar graphs, with a reference to [1]. However, the problem solved in that publication, which we refer to as the planar minLA, is different from the minLA, as we show in this article. In constrast to the minimum linear arrangement problem (minLA), the planar minimum linear arrangement ...

The crossing number of Kn is known for n 6 10. We develop several simple counting properties that we shall exploit in showing by computer that cr(K11) = 100, which implies that cr(K12) = 150. We also determine the numbers of non-isomorphic optimal drawings of K9 and K10.

Crossing number bounds for the mesh of trees graph are derived.