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  2. Crossing number inequality - Wikipedia

    en.wikipedia.org/wiki/Crossing_number_inequality

    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]

  3. Crossing number (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Crossing_number_(graph_theory)

    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.

  4. Crossing Numbers of Graphs - Wikipedia

    en.wikipedia.org/wiki/Crossing_Numbers_of_Graphs

    [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.

  5. Turán's brick factory problem - Wikipedia

    en.wikipedia.org/wiki/Turán's_brick_factory_problem

    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]

  6. Szemerédi–Trotter theorem - Wikipedia

    en.wikipedia.org/wiki/Szemerédi–Trotter_theorem

    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.

  7. Albertson conjecture - Wikipedia

    en.wikipedia.org/wiki/Albertson_conjecture

    It is straightforward to show that graphs with bounded crossing number have bounded chromatic number: one may assign distinct colors to the endpoints of all crossing edges and then 4-color the remaining planar graph. Albertson's conjecture replaces this qualitative relationship between crossing number and coloring by a more precise quantitative ...

  8. Graph theory - Wikipedia

    en.wikipedia.org/wiki/Graph_theory

    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 ...

  9. Three utilities problem - Wikipedia

    en.wikipedia.org/wiki/Three_utilities_problem

    The question of minimizing the number of crossings in drawings of complete bipartite graphs is known as Turán's brick factory problem, and for , the minimum number of crossings is one. K 3 , 3 {\displaystyle K_{3,3}} is a graph with six vertices and nine edges, often referred to as the utility graph in reference to the problem. [ 1 ]