We give a natural sufficient condition for an intersection graph of compact convex sets in R d to have a balanced separator of sublinear size. This condition generalizes several previous results on sublinear separators in intersection graphs. Furthermore, the argument used to prove the existence of sublinear separators is based on a connection with generalized coloring numbers which has not been previously explored in geometric settings.
A graph G is k-ordered if for any distinct vertices v 1 , v 2 , . . . , v k ∈ V (G), it has a cycle through v 1 , v 2 , . . . , v k in order. Let f (k) denote the minimum integer so that every f (k)-connected graph is k-ordered. The first non-trivial case of determining f (k) is when k = 4, where the previously best known bounds are 7 ≤ f (4) ≤ 40. We prove that in fact f (4) = 7.
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