Counting paths in digraphs
Combinatorics
2012-11-02 v1
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 We present an unpublished result of Bondy proving that there are at most 2n^3/25 such paths, and prove that for the class of circular interval digraphs, an upper bound of n^3/16 holds.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.
Cite
@article{arxiv.1210.8424,
title = {Counting paths in digraphs},
author = {Paul Seymour and Blair D. Sullivan},
journal= {arXiv preprint arXiv:1210.8424},
year = {2012}
}