Applications of a new separator theorem for string graphs

TL;DR

This paper explores new separator theorems for string graphs and demonstrates their implications for graph coloring, edge bounds, and Ramsey theory, advancing understanding of string graph structure and properties.

Contribution

It combines separator bounds with dense string graph properties to derive new results on coloring, edge limits, and Ramsey-type phenomena in string graphs.

Findings

String graphs of m edges have separators of size O(√m log m).

Dense string graphs contain large complete bipartite subgraphs.

Results imply bounds on chromatic number and edges for string graphs excluding certain subgraphs.

Abstract

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if K_t is not a subgraph of G for some t, then the chromatic number of G is at most (\log n)^{O(\log t)}; (ii) if K_{t,t} is not a subgraph of G, then G has at most t(\log t)^{O(1)}n edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds for string graphs.