Directed Graphs Without Short Cycles

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)/r, 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 < θ < 1/2 and sufficiently large n, if G is a digraph with n vertices and β(G) ≥ θn, then for any 0 ≤ m ≤ θn− o(n) it contains a directed cycle whose length is between m and m+ 6θ. 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.