A note on digraph splitting
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| Název: | A note on digraph splitting |
|---|---|
| Autoři: | Micha Christoph, Kalina Petrova, Raphael Steiner |
| Zdroj: | Combinatorics, Probability and Computing. 34:559-564 |
| Publication Status: | Preprint |
| Informace o vydavateli: | Cambridge University Press (CUP), 2025. |
| Rok vydání: | 2025 |
| Témata: | Combinatorics, splitting, FOS: Mathematics, Digraph, minimum out-degree, Combinatorics (math.CO), directed graphs |
| Popis: | A tantalizing open problem, posed independently by Stiebitz in 1995 and by Alon in 1996 and again in 2006, asks whether for every pair of integers $s,t \ge 1$ there exists a finite number $F(s,t)$ such that the vertex set of every digraph of minimum out-degree at least $F(s,t)$ can be partitioned into non-empty parts $A$ and $B$ such that the subdigraphs induced on $A$ and $B$ have minimum out-degree at least $s$ and $t$ , respectively.In this short note, we prove that if $F(2,2)$ exists, then all the numbers $F(s,t)$ with $s,t\ge 1$ exist and satisfy $F(s,t)=\Theta (s+t)$ . In consequence, the problem of Alon and Stiebitz reduces to the case $s=t=2$ . Moreover, the numbers $F(s,t)$ with $s,t \ge 2$ either all exist and grow linearly, or all of them do not exist. |
| Druh dokumentu: | Article |
| Jazyk: | English |
| ISSN: | 1469-2163 0963-5483 |
| DOI: | 10.1017/s0963548325000045 |
| DOI: | 10.3929/ethz-b-000728804 |
| DOI: | 10.48550/arxiv.2310.08449 |
| Přístupová URL adresa: | http://arxiv.org/abs/2310.08449 |
| Rights: | CC BY arXiv Non-Exclusive Distribution |
| Přístupové číslo: | edsair.doi.dedup.....f4d4b2b3689cfa580c6eb41a56549431 |
| Databáze: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | A tantalizing open problem, posed independently by Stiebitz in 1995 and by Alon in 1996 and again in 2006, asks whether for every pair of integers $s,t \ge 1$ there exists a finite number $F(s,t)$ such that the vertex set of every digraph of minimum out-degree at least $F(s,t)$ can be partitioned into non-empty parts $A$ and $B$ such that the subdigraphs induced on $A$ and $B$ have minimum out-degree at least $s$ and $t$ , respectively.In this short note, we prove that if $F(2,2)$ exists, then all the numbers $F(s,t)$ with $s,t\ge 1$ exist and satisfy $F(s,t)=\Theta (s+t)$ . In consequence, the problem of Alon and Stiebitz reduces to the case $s=t=2$ . Moreover, the numbers $F(s,t)$ with $s,t \ge 2$ either all exist and grow linearly, or all of them do not exist. |
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| ISSN: | 14692163 09635483 |
| DOI: | 10.1017/s0963548325000045 |