Applications of a New Separator Theorem for String Graphs
2013 ◽
Vol 23
(1)
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pp. 66-74
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Keyword(s):
An intersection graph of curves in the plane is called astring graph. Matoušek almost completely settled a conjecture of the authors by showing that every string graph withmedges 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 graphsGwithnvertices: (i) ifKt⊈Gfor somet, then the chromatic number ofGis at most (logn)O(logt); (ii) ifKt,t⊈G, thenGhas at mostt(logt)O(1)nedges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdős–Hajnal conjecture almost holds for string graphs.
2009 ◽
Vol 19
(3)
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pp. 371-390
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Keyword(s):
2013 ◽
Vol 23
(1)
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pp. 135-139
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2012 ◽
Vol 11
(01)
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pp. 1250019
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Keyword(s):
Keyword(s):
1970 ◽
Vol 22
(5)
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pp. 1082-1096
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2019 ◽
Vol 101
(3)
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pp. 362-366
Keyword(s):
2016 ◽
Vol 99
(10)
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pp. 1571--1582
2015 ◽
Vol 07
(04)
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pp. 1550040
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Keyword(s):