We compute the Orchard crossing number, which is defined in a similar way to the rectilinear crossing number, for the complete bipartite graphs Kn,n.

Let cr(Kn) be the minimum number of crossings over all rectilinear drawings of the complete graph on n vertices on the plane. In this paper we prove that cr(Kn) < 0.380473 (

For a torus knot K, we bound the crosscap number c(K) in terms of the genus g(K) and crossing number n(K): c(K) ≤ ⌊(g(K) + 9)/6⌋ and c(K) ≤ ⌊(n(K)+16)/12⌋. The (6n− 2, 3) torus knots show that these bounds are sharp.

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.

We show that the edges crossed by a random walk in a network form a recurrent graph a.s. In fact, the same is true when those edges are weighted by the number of crossings. §

We give an algorithm that decides whether the bipartite crossing number of a given graph is at most k. The running time of the algorithm is upper bounded by 2 + n, where n is the number of vertices of the input graph, which improves the previously known algorithm due to Kobayashi et al. (TCS 2014) that runs in 2 log +n time. This result is based on a combinatorial upper bound on the number of t...

iráň constructed infinite families of k-crossing-critical graphs for every k ≥ 3 and Kochol constructed such families of simple graphs for every k ≥ 2. Richter and Thomassen argued that, for any given k ≥ 1 and r ≥ 6, there are only finitely many simple k-crossingcritical graphs with minimum degree r. Salazar observed that the same argument implies such a conclusion for simple k-crossing-critic...

We prove that any non-hyperbolic genus one knot except the trefoil does not have a minimal canonical Seifert surface and that there are only polynomially many in the crossing number positive knots of given genus or given unknotting number.

In this paper, we study the crossing number of the complete bipartite graph K4,n in torus and obtain crT (K4,n) = ⌊ n 4 ⌋(2n− 4(1 + ⌊ n 4 ⌋)).

The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We show that many families of alternating knots and links simultaneously maximi...