Applications of a New Separator Theorem for String Graphs

Abstract

An intersection graph of curves in the plane is called a string graph. Matoušek almost completely settled a conjecture of the authors by showing that every string graph with 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 Kt ⊈ G for some t, then the chromatic number of G is at most (log n) O(log t); (ii) if Kt,t ⊈ G, then G has at most t(log t) O(1) n edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdős-Hajnal conjecture almost holds for string graph