Applications of a New Separator Theorem for String Graphs
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Fox_Applications of.pdf
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Author(s)
Fox, Jacob
Pach, Janos
Date Issued
October 2013
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
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1017/S0963548313000412
