In this note we give a simple proof of a result of Richter and Siran by basic counting method, which says that the crossing number of K3,n in a surface with Euler genus ε is

We prove that the number γN of the zeros of a two-parameter simple random walk in its first N × N time steps is almost surely equal to N as N →∞. This is in contrast with our earlier joint effort with Z. Shi [4]; that work shows that the number of zero crossings in the first N × N time steps is N (3/2)+o(1) as N → ∞. We prove also that the number of zeros on the diagonal in the first N time ste...

The crossing number of Sierpiński graphs S(n, k) and their regularizations S(n, k) and S(n, k) is studied. Explicit drawings of these graphs are presented and proved to be optimal for S(n, k) and S(n, k) for every n ≥ 1 and k ≥ 1. These are the first nontrivial families of graphs of “fractal” type whose crossing number is known.

In 1958, Hill conjectured that the minimum number of crossings in a drawing of Kn is exactly Z(n) = 1 4 ⌊n 2 ⌋ ⌊

The biplanar crossing number cr2(G) of a graph G is min{cr(G1)+ cr(G2)}, where cr is the planar crossing number and G1 ∪ G2 = G. We show that cr2(G) ≤ (3/8)cr(G). Using this result recursively, we bound the thickness by Θ(G) − 2 ≤ Kcr2(G) log2 n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time randomized algorithm. We show that for an...

A naive but often useful way of thinking of a knot or link in R3 is to generically project it onto a plane, keeping account of which part of the knot goes under and which part goes over at any given crossing. Think of laying the knot on a table and taking note of how it crosses itself whenever one part of it lies on top of another. It’s natural to ask how the knot is changed by altering one of ...

This section offers a series of joint reflections on (open) data platform from a variety of cases, from cycling, traffic and mapping to activism, environment and data brokering. Data platforms play a key role in contemporary urban governance. Linked to open data initiatives, such platforms are often proposed as both mechanisms for enhancing the accountability of administrations and performing a...

The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NP-complete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations.

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...