Coloring copoints of a planar point set
نویسنده
چکیده
To a set of n points in the plane, one can associate a graph that has less than n2 vertices and has the property that k-cliques in the graph correspond vertex sets of convex k-gons in the point set. We prove an upper bound of 2k−1 on the size of a planar point set for which the graph has chromatic number k, matching the bound conjectured by Szekeres for the clique number. Constructions of Erdős and Szekeres are shown to yield graphs that have very low chromatic number. The constructions are carried out in the context of pseudoline arrangements.
منابع مشابه
Points, Copoints, and Colorings
In 1935, Paul Erdős and George Szekeres were able to show that any point set large enough contains the vertices of a convex k-gon. Later in 1961, they constructed a point set of size 2k−2 not containing the vertex set of any convex k-gon. This leads to what is known as the Erdős-Szekeres Conjecture, that any point set of 2k−2 + 1 points contains the vertices of a convex k-gon. Recently, this fa...
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ورودعنوان ژورنال:
- Discrete Applied Mathematics
دوره 154 شماره
صفحات -
تاریخ انتشار 2006