## Applications of a New Separator Theorem for String Graphs

##### Author(s)

Fox, Jacob; Pach, Janos
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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.

##### Date issued

2013-10##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Combinatorics, Probability and Computing

##### Publisher

Cambridge University Press

##### Citation

Fox, Jacob, and Janos Pach. “Applications of a New Separator Theorem for String Graphs.” Combinatorics, Probability and Computing 23, no. 01 (January 2014): 66–74.

Version: Original manuscript

##### ISSN

0963-5483

1469-2163