Directed graphs without short cycles

  • Authors:

  • Jacob Fox;Peter Keevash;Benny Sudakov

  • Affiliations:

  • Department of mathematics, princeton, nj 08544, usa (e-mail: jacobfox@math.princeton.edu);School of mathematical sciences, queen mary, university of london, mile end road, london e1 4ns, uk (e-mail: p.keevash@qmul.ac.uk);Department of mathematics, ucla, los angeles, ca 90095, usa (e-mail: bsudakov@math.ucla.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2010

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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 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.