Low chromatic spanning sub(di)graphs with prescribed degree or connectivity properties
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| Titel: | Low chromatic spanning sub(di)graphs with prescribed degree or connectivity properties |
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| Autoren: | Anders Yeo, Frédéric Havet, Joergen Bang-Jensen, Matthias Kriesell |
| Weitere Verfasser: | Havet, Frederic |
| Quelle: | Bang-Jensen, J, Havet, F, Kriesell, M & Yeo, A 2022, ' Low chromatic spanning sub(di)graphs with prescribed degree or connectivity properties ', Journal of Graph Theory, vol. 99, no. 4, pp. 615-636 . https://doi.org/10.1002/jgt.22755 |
| Publication Status: | Preprint |
| Verlagsinformationen: | Wiley, 2021. |
| Publikationsjahr: | 2021 |
| Schlagwörter: | strong connectivity, 0102 computer and information sciences, semicomplete digraph, 01 natural sciences, 05C20, 05C15, edge-disjoint spanning trees, [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], bipartite graph, FOS: Mathematics, Mathematics - Combinatorics, arc-connectivity, Combinatorics (math.CO), majority colouring, 0101 mathematics, edge-connectivity |
| Beschreibung: | Generalizing well‐known results of Erdős and Lovász, we show that every graph contains a spanning ‐partite subgraph with , where is the edge‐connectivity of . In particular, together with a well‐known result due to Nash‐Williams and Tutte, this implies that every 7‐edge‐connected graph contains a spanning bipartite graph whose edge set decomposes into two edge‐disjoint spanning trees. We show that this is best possible as it does not hold for infinitely many 6‐edge‐connected graphs. For directed graphs, it was shown by Bang‐Jensen et al. that there is no such that every ‐arc‐connected digraph has a spanning strong bipartite subdigraph. We prove that every strong digraph has a spanning strong 3‐partite subdigraph and that every strong semicomplete digraph on at least six vertices contains a spanning strong bipartite subdigraph. We generalize this result to higher connectivities by proving that, for every positive integer , every ‐arc‐connected digraph contains a spanning ()‐partite subdigraph which is ‐arc‐connected and this is best possible. A conjecture by Kreutzer et al. implies that every digraph of minimum out‐degree contains a spanning 3‐partite subdigraph with minimum out‐degree at least . We prove that the bound would be best possible by providing an infinite class of digraphs with minimum out‐degree which do not contain any spanning 3‐partite subdigraph in which all out‐degrees are at least . We also prove that every digraph of minimum semidegree at least contains a spanning 6‐partite subdigraph in which every vertex has in‐ and out‐degree at least . |
| Publikationsart: | Article |
| Sprache: | English |
| ISSN: | 1097-0118 0364-9024 |
| DOI: | 10.1002/jgt.22755 |
| DOI: | 10.48550/arxiv.2008.05272 |
| Zugangs-URL: | http://arxiv.org/pdf/2008.05272 http://arxiv.org/abs/2008.05272 https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.22755 https://portal.findresearcher.sdu.dk/en/publications/low-chromatic-spanning-subdigraphs-with-prescribed-degree-or-conn https://portal.findresearcher.sdu.dk/da/publications/36a921dd-0e16-43c0-8baa-5725ab820e0a https://doi.org/10.1002/jgt.22755 https://portal.findresearcher.sdu.dk/da/publications/36a921dd-0e16-43c0-8baa-5725ab820e0a |
| Rights: | Wiley Online Library User Agreement arXiv Non-Exclusive Distribution |
| Dokumentencode: | edsair.doi.dedup.....abfef6a80d56b00ee52e0659962476eb |
| Datenbank: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | Generalizing well‐known results of Erdős and Lovász, we show that every graph contains a spanning ‐partite subgraph with , where is the edge‐connectivity of . In particular, together with a well‐known result due to Nash‐Williams and Tutte, this implies that every 7‐edge‐connected graph contains a spanning bipartite graph whose edge set decomposes into two edge‐disjoint spanning trees. We show that this is best possible as it does not hold for infinitely many 6‐edge‐connected graphs. For directed graphs, it was shown by Bang‐Jensen et al. that there is no such that every ‐arc‐connected digraph has a spanning strong bipartite subdigraph. We prove that every strong digraph has a spanning strong 3‐partite subdigraph and that every strong semicomplete digraph on at least six vertices contains a spanning strong bipartite subdigraph. We generalize this result to higher connectivities by proving that, for every positive integer , every ‐arc‐connected digraph contains a spanning ()‐partite subdigraph which is ‐arc‐connected and this is best possible. A conjecture by Kreutzer et al. implies that every digraph of minimum out‐degree contains a spanning 3‐partite subdigraph with minimum out‐degree at least . We prove that the bound would be best possible by providing an infinite class of digraphs with minimum out‐degree which do not contain any spanning 3‐partite subdigraph in which all out‐degrees are at least . We also prove that every digraph of minimum semidegree at least contains a spanning 6‐partite subdigraph in which every vertex has in‐ and out‐degree at least . |
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| ISSN: | 10970118 03649024 |
| DOI: | 10.1002/jgt.22755 |