Abstract
AbstractA string graph is the intersection graph of curves in the plane. We prove that for every $$\epsilon >0$$
ϵ
>
0
, if G is a string graph with n vertices such that the edge density of G is below $${1}/{4}-\epsilon $$
1
/
4
-
ϵ
, then V(G) contains two linear sized subsets A and B with no edges between them. The constant 1/4 is a sharp threshold for this phenomenon as there are string graphs with edge density less than $${1}/{4}+\epsilon $$
1
/
4
+
ϵ
such that there is an edge connecting any two logarithmic sized subsets of the vertices. The existence of linear sized sets A and B with no edges between them in sufficiently sparse string graphs is a direct consequence of a recent result of Lee about separators. Our main theorem finds the largest possible density for which this still holds. In the special case when the curves are x-monotone, the same result was proved by Pach and the author of this paper, who also proposed the conjecture for the general case.
Funder
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
Ministry of Education and Science of the Russian Federation
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Theoretical Computer Science
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