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Applications of a New Separator Theorem for String Graphs

Published online by Cambridge University Press:  25 October 2013

JACOB FOX  and
JÁNOS PACH
JACOB FOX
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA (e-mail: fox@math.mit.edu)
JÁNOS PACH
Affiliation:
École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland and Rényi Institute, Hungarian Academy of Sciences, PO Box 127, H-1364 Budapest, Hungary (e-mail: pach@cims.nyu.edu)
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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 graphs.

Keywords

Primary 05C35Secondary 05C6252C10

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 1 , January 2014 , pp. 66 - 74
Copyright
Copyright © Cambridge University Press 2013 

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