The minor crossing number of a graph G, mcr(G), is defined as the minimum crossing number of all graphs that contain G as a minor. We present some basic properties of this new minor-monotone graph invariant. We give estimates on mcr for some important graph families using the topological structure of graphs satisfying mcr(G) ≤ k.

We define the crossing number, a natural invariant in topological graph theory, and survey general results on the crossing number in the Euclidean plane. We also discuss progress on conjectures for two particular cases: complete bipartite graphs and complete graphs. Finally, we show some applications of the crossing number to problems in incidence geometry and VLSI chip design.

The problem of drawing a graph in the plane with a minimum number of edge crossings—called the crossing number of a graph—is a well-studied problem which dates back to the first half of the twentieth century, as mentioned in [11], and was formulated in full generality in [3]. It was shown that this problem is NP-Complete [4], and that it remains so even when restricted to cubic graphs [5]. Many...

In 1940, the Hungarian mathematician Paul Turán was sent to a forced labor camp by the Nazis. Though every part of his life was brutally controlled, he still managed to do serious mathematics under the most extreme conditions. While forced to collect wire from former neighborhoods, he would be thinking about mathematics. When he found scraps of paper, he wrote down his theorems and conjectures....

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...