Graphs drawn with few crossings per edge

  title={Graphs drawn with few crossings per edge},
  author={J{\'a}nos Pach and G{\'e}za T{\'o}th},
We show that if a graph ofv vertices can be drawn in the plane so that every edge crosses at mostk>0 others, then its number of edges cannot exceed 4.108√kv. Fork≤4, we establish a better bound, (k+3)(v−2), which is tight fork=1 and 2. We apply these estimates to improve a result of Ajtai et al. and Leighton, providing a general lower bound for the crossing number of a graph in terms of its number of vertices and edges. 
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Let G be a graph that can be drawn on a plane in such a way that any edge intersects at most one other edge. It is proved that the chromatic number of G does not exceed 7. The bound $ \chi (G)\leq
A note on 1-planar graphs
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  • L. Székely
  • Mathematics
    Combinatorics, Probability and Computing
  • 1997
We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the
Combinatorial complexity bounds for arrangements of curves and spheres
Upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike are presented and it is proved that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m2/3n 2/3 +n), and that it isO(m3/2β(m) forn unit-circles.
Extremal problems in discrete geometry
Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.
On the Number of Incidences Between Points and Curves
  • J. Pach, M. Sharir
  • Mathematics, Economics
    Combinatorics, Probability and Computing
  • 1998
We apply an idea of Székely to prove a general upper bound on the number of incidences between a set of m points and a set of n ‘well-behaved’ curves in the plane.
Combinatorial geometry
  • J. Pach, P. Agarwal
  • Mathematics
    Wiley-Interscience series in discrete mathematics and optimization
  • 1995
An Introduction to the Theory of Numbers
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Complexity Issues in VLSI