Counting paths in digraphs

Paul D. Seymour, Blair D. Sullivan

Abstract: Say a digraph is k<math><mi is="true">k</mi></math>-free if it has no directed cycles of length at most k<math><mi is="true">k</mi></math>, for k∈Z+<math><mi is="true">k</mi><mo is="true">∈</mo><msup is="true"><mrow is="true"><mi mathvariant="double-struck" is="true">Z</mi></mrow><mrow is="true"><mo is="true">+</mo></mrow></msup></math>. Thomassé conjectured that the number of induced 3-vertex directed paths in a simple 2-free digraph on n<math><mi is="true">n</mi></math> vertices is at most (n−1)n(n+1)/15<math><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">−</mo><mn is="true">1</mn><mo is="true">)</mo></mrow><mi is="true">n</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">+</mo><mn is="true">1</mn><mo is="true">)</mo></mrow><mo is="true">/</mo><mn is="true">15</mn></math>. We present an unpublished result of Bondy proving that there are at most 2n3/25<math><mn is="true">2</mn><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup><mo is="true">/</mo><mn is="true">25</mn></math> such paths, and prove that for the class of circular interval digraphs, an upper bound of n3/16<math><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup><mo is="true">/</mo><mn is="true">16</mn></math> 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 4n4/75<math><mn is="true">4</mn><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup><mo is="true">/</mo><mn is="true">75</mn></math> using Bondy’s result for Thomassé’s conjecture.