Abstract
A string graph is the intersection graph of curves in the plane. We prove that for every epsilon >0, if G is a string graph with n vertices such that the edge density of G is below {1}/{4}-epsilon , 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 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.
Highlights
The intersection graph of a family of sets C is the graph whose vertices are identified with the elements of C, and two vertices are joined by an edge if the corresponding elements of C have a non-empty intersection
Pach and Tomon [16] proved that if we restrict our attention to x-monotone curves, there is a sharp threshold for the edge density when linear sized bi-cliques start to appear in G. They proved that for every > 0 there exists δ > 0 such that if G is the intersection graph of n x-monotone curves and |E(G)| ≤ (1/4 − )n2/2, G contains a bi-clique of size at least δn
The existence of linear sized bi-cliques in the complement of not too dense string graphs follows from a recent graph theoretic result of Chudnovsky et al [2]: Theorem 1.3 For every graph H, there exists > 0 such that if G is a graph with n vertices and at most n2 edges, and G does not contain a subdivision of H as an induced subgraph, G contains a bi-clique of size at least n
Summary
The intersection graph of a family of sets C is the graph whose vertices are identified with the elements of C , and two vertices are joined by an edge if the corresponding elements of C have a non-empty intersection. They conjectured a strengthening of their result that the same conclusion holds if m denotes the number of intersecting pairs of curves in C Almost settling this conjecture, Matouš√ek [14] proved that every string graph G with m edges contains a separator of size O( m log m). The existence of linear sized bi-cliques in the complement of not too dense string graphs follows from a recent graph theoretic result of Chudnovsky et al [2]: Theorem 1.3 For every graph H , there exists > 0 such that if G is a graph with n vertices and at most n2 edges, and G does not contain a subdivision of H as an induced subgraph, G contains a bi-clique of size at least n.
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