Applications of a New Separator Theorem for String Graphs

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 with m edges admits a vertex separator of size O(√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[subscript t] ⊈ G for some t, then the chromatic number of G is at most (log n) [superscript O(log t)]; (ii) if K[subscript t,t] ⊈ G, then G has at most t(log t) [superscript O(1)] n edges; and (iii) a lopsided Ramsey-type result, which shows that the Erdos–Hajnal conjecture almost holds for string graphs.