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Directed Graphs Without Short Cycles

Published online by Cambridge University Press:  01 October 2009

JACOB FOX ,
PETER KEEVASH  and
BENNY SUDAKOV
JACOB FOX
Affiliation:
Department of Mathematics, Princeton, NJ 08544, USA (e-mail: jacobfox@math.princeton.edu)
PETER KEEVASH
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK (e-mail: p.keevash@qmul.ac.uk)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA (e-mail: bsudakov@math.ucla.edu)
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Abstract

For a directed graph G without loops or parallel edges, let β(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 γ(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 β(G) ≤ cγ(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 also be used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed 0 < θ < 1/2 and sufficiently large n, if G is a digraph with n vertices and β(G) ≥ θn2, then for any 0 ≤ m ≤ θn − o(n) it contains a directed cycle whose length is between m and m + 6θ−1/2. Moreover, there is a constant C such that either G contains directed cycles of every length between C and θn − o(n) or it is close to a digraph G′ with a simple structure: every strong component of G′ is periodic. These results are also tight up to the constant factors.

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 2 , March 2010 , pp. 285 - 301
Copyright
Copyright © Cambridge University Press 2009

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