On the length of directed paths in digraphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913474884403200 |
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| author | Cheng, Yangyang Keevash, Peter |
| author_facet | Cheng, Yangyang Keevash, Peter |
| contents | Thomassé conjectured the following strengthening of the well-known Caccetta-Haggkvist Conjecture: any digraph with minimum out-degree $δ$ and girth $g$ contains a directed path of length $δ(g-1)$. Bai and Manoussakis \cite{Bai} gave counterexamples to Thomassé's conjecture for every even $g\geq 4$. In this note, we first generalize their counterexamples to show that Thomassé's conjecture is false for every $g\geq 4$. We also obtain the positive result that any digraph with minimum out-degree $δ$ and girth $g$ contains a directed path of $2(1-\frac{2}{g})$. For small $g$ we obtain better bounds, e.g.~for $g=3$ we show that oriented graph with minimum out-degree $δ$ contains a directed path of length $1.5δ$. Furthermore, we show that each $d$-regular digraph with girth $g$ contains a directed path of length $Ω(dg/\log d)$. Our results give the first non-trivial bounds for these problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16776 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the length of directed paths in digraphs Cheng, Yangyang Keevash, Peter Combinatorics Thomassé conjectured the following strengthening of the well-known Caccetta-Haggkvist Conjecture: any digraph with minimum out-degree $δ$ and girth $g$ contains a directed path of length $δ(g-1)$. Bai and Manoussakis \cite{Bai} gave counterexamples to Thomassé's conjecture for every even $g\geq 4$. In this note, we first generalize their counterexamples to show that Thomassé's conjecture is false for every $g\geq 4$. We also obtain the positive result that any digraph with minimum out-degree $δ$ and girth $g$ contains a directed path of $2(1-\frac{2}{g})$. For small $g$ we obtain better bounds, e.g.~for $g=3$ we show that oriented graph with minimum out-degree $δ$ contains a directed path of length $1.5δ$. Furthermore, we show that each $d$-regular digraph with girth $g$ contains a directed path of length $Ω(dg/\log d)$. Our results give the first non-trivial bounds for these problems. |
| title | On the length of directed paths in digraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.16776 |