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How to draw a planar graph on a grid
It is shown that any setF, which can support a Fáry embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.
Research problems in discrete geometry
This chapter discusses 100 Research Problems in Discrete Geometry from the Facsimile edition of the World Classics in Mathematics Series, vol.
On the Number of Incidences Between Points and Curves
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.
Graphs drawn with few crossings per edge
It is shown 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, and a better bound is established, (k+3)(v−2), which is tight fork=1 and 2.
Repeated Angles in the Plane and Related Problems
Wiley‐Interscience Series in Discrete Mathematics and Optimization
Embedding Planar Graphs at Fixed Vertex Locations
An algorithm for constructing in O(n2) time a planar embedding of G, where vertices are represented by point pi and each edge is represented by a polygonal curve with O( n) bends (internal vertices).
Forbidden paths and cycles in ordered graphs and matrices
At most how many edges can an ordered graph ofn vertices have if it does not contain a fixed forbidden ordered subgraphH? It is not hard to give an asymptotically tight answer to this question,…
Conflict-Free Colourings of Graphs and Hypergraphs
An efficient deterministic algorithm is given to find a conflict-free colouring of the vertices of a hypergraph H if each hyperedge E of H contains a vertex of ‘unique’ colour that does not get repeated in E, and the smallest number of colours required is denoted by χCF(H).