Almost Given Length Cycles in Digraphs

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Almost Given Length Cycles in Digraphs
For a directed graph G without loops or parallel edges, let (G) denote the size of the smallest feedback arc set, i.e., the smallest subset X E(G) such that G \ X has no directed cycles. Let (G) be the number of unordered pairs of vertices of G which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least r 4 satisfies (G) c(G)/r2 , where c is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour, and Sullivan. This result can be also used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed 0 < < 1/2 and sufficiently large n, if G is a digraph with n vertices and (G) n2 , then for any 0 m n - o(n) it contains a directed cycle whose length is between m and m + 6-1/2 . Moreover, there is a constant C such that either G contains directed cycles of every length between C and n - o(n) or it is close to a digraph G with a si...
Raphael Yuster
Added 10 Dec 2010
Updated 10 Dec 2010
Type Journal
Year 2008
Where GC
Authors Raphael Yuster
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