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The crossing number inequality states that, for an undirected simple graph G with n vertices and e edges such that e > 7n, the crossing number cr(G) obeys the inequality (). The constant 29 is the best known to date, and is due to Ackerman. [3]
Turán's conjectured formula for the crossing numbers of complete bipartite graphs remains unproven, as does an analogous formula for the complete graphs. The crossing number inequality states that, for graphs where the number e of edges is sufficiently larger than the number n of vertices, the crossing number is at least proportional to e 3 /n 2.
[2] [3] It also includes the crossing number inequality, and the Hanani–Tutte theorem on the parity of crossings. [1] The second chapter concerns other special classes of graphs including graph products (especially products of cycle graphs) and hypercube graphs.
Since each line segment lies on one of m lines, and any two lines intersect in at most one point, the crossing number of this graph is at most the number of points where two lines intersect, which is at most m(m − 1)/2. The crossing number inequality implies that either e ≤ 7.5n, or that m(m − 1)/2 ≥ e 3 / 33.75n 2.
A simple proof of this follows from the crossing number inequality: [15] if cells have a total of + edges, one can form a graph with nodes (one per cell) and edges (one per pair of consecutive cells on the same line). The edges of this graph can be drawn as curves that do not cross within the cells corresponding to their endpoints, and then ...
A crossing is counted whenever two edges that are disjoint in the graph have a nonempty intersection in the plane. The question is then, what is the minimum number of crossings in such a drawing? [2] [3] Turán's formulation of this problem is often recognized as one of the first studies of the crossing numbers of graphs. [4]
With Chvátal, Newborn, and Szemerédi, Ajtai proved the crossing number inequality, that any drawing of a graph with n vertices and m edges, where m > 4n, has at least m 3 / 100n 2 crossings. Ajtai and Dwork devised in 1997 a lattice-based public-key cryptosystem; Ajtai has done extensive work on lattice problems. For his numerous ...
Graph drawing also can be said to encompass problems that deal with the crossing number and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a planar graph, the crossing number is zero by definition. Drawings on surfaces ...