For r = 7, 8 we prove that if χ(G) = r, then cr(G) ≥ cr(Kr). The case r = 5 of the Weak Hajós Conjecture is equivalent to the Four Color Theorem, and the case r = 6 follows from a theorem of Oporowski and Zhao. We present partial results when r = 9.

We prove tight crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of crossing-free geometric graphs that can be drawn on such grids. In particular, we show that any geometric graph with m ≥ 8N edges and with vertices on a 3D integer grid of volumeN , has Ω((m2/n) l...

The River and Stream Continuity Partnership has developed assessment protocols for evaluating the barrier effects of road-stream crossings. In addition to these protocols, the University of Massachusetts Amherst created an online database and an algorithm for scoring crossing structures according to the degree of obstruction they pose to aquatic organisms. In 2008, The Nature Conservancy and Un...

The use of null hypotheses (in a statistical sense) is common in hard sciences but not in theoretical linguistics. Here the null hypothesis that the low frequency of syntactic dependency crossings is expected by an arbitrary ordering of words is rejected. It is shown that this would require star dependency structures, which are both unrealistic and too restrictive. The hypothesis of the limited...

In many physical systems it is important to be aware of the crossings and avoided crossings which occur when eigenvalues of a physical observable are varied using an external parameter. We have discovered a powerful algebraic method of finding such crossings via a mapping to the problem of locating the roots of a polynomial in that parameter. We demonstrate our method on atoms and molecules in ...

This paper proposes a gradient ascent learning algorithm of the Hopfield neural networks for solving fixed linear crossing number problem. The fixed linear crossing number problem is an important problem in printed circuit board layout, VLSI circuit routing, and automated graph drawing. The objective of this problem which is shown to be NP-hard is to embed the edges so that the total number of ...

Let G be a graph on n vertices; let S be a set of n points in general position (no three of them collinear) in the plane. A rectilinear drawing of G is a drawing of G in the plane that satisfies the following. Its vertices are points in general position and its edges are drawn as straight line segments. The number of crossings of a rectilinear drawing is the number of pairs of its edges that cr...

In a drawing of a graph, two edges form an odd pair if they cross each other an odd number of times. A pair of edges is independent if they share no endpoint. For a graph G, let ocr(G) be the smallest number of odd pairs in a drawing of G and let iocr(G) be the smallest number of independent odd pairs in a drawing of G. We construct a graph G with iocr(G) < ocr(G), answering a question by Széke...

We show that cr(G) ≤ (2 iocr(G) 2 ) settling an open problem of Pach and Tóth [5,1]. Moreover, iocr(G) = cr(G) if iocr(G) ≤ 2.

Every year, more than 400 people are killed in over 1200 accidents at road-rail level crossings in the European Union. Together with tunnels and specific road black spots, level crossings (LC) have been identified as being a particular weak point in road infrastructure, seriously jeopardizing road safety. Safety at LC is a single part of a wider picture of transport safety within the whole tran...