Counting paths in digraphs

TL;DR

This paper investigates bounds on the number of specific directed paths in k-free digraphs, providing new upper bounds for 3-vertex and 4-vertex paths in various classes of digraphs.

Contribution

It proves new upper bounds for the number of paths in k-free digraphs, including an unpublished result and bounds specific to circular interval digraphs.

Findings

At most 2n^3/25 induced 3-vertex paths in 2-free digraphs.

In circular interval digraphs, at most n^3/16 such paths.

Bound of 4n^4/75 on 4-vertex paths in 3-free digraphs.

Abstract

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 We present an unpublished result of Bondy proving that there are at most 2n^3/25 such paths, and prove that for the class of circular interval digraphs, an upper bound of n^3/16 holds.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.