Counting paths in digraphs

Say a digraph is k -free if it has no directed cycles of length at most k , for k ∈ Z + . Thomassé conjectured that the number of induced 3-vertex directed paths in a simple 2-free digraph on n vertices is at most ( n − 1 ) n ( n + 1 ) / 15 . We present an unpublished result of Bondy proving that there are at most 2 n 3 / 25 such paths, and prove that for the class of circular interval digraphs, an upper bound of n 3 / 16 holds. We also study the problem of bounding the number of (non-induced) 4-vertex paths in 3-free digraphs. We show an upper bound of 4 n 4 / 75 using Bondy’s result for Thomassé’s conjecture.