4 research outputs found
Counting Paths in Digraphs
Say a digraph is k-free if it has no directed cycles of length at most k, for
positive integers k. Thomasse 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 We present an unpublished result of Bondy proving that there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 2n^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 4n^4/75 using Bondy's result for Thomasse's conjecture