Abstract
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 We present an unpublished result of Bondy proving that there are at most 2 n3 / 25 such paths, and prove that for the class of circular interval digraphs, an upper bound of n3 / 16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 n4 / 75 using Bondy's result for Thomassé's conjecture.
Original language | English (US) |
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Pages (from-to) | 961-975 |
Number of pages | 15 |
Journal | European Journal of Combinatorics |
Volume | 31 |
Issue number | 3 |
DOIs | |
State | Published - Apr 2010 |
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics