Star-Forest Decompositions of Complete Graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929246431084544 |
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| author | Antić, Todor Glišić, Jelena Milivojčević, Milan |
| author_facet | Antić, Todor Glišić, Jelena Milivojčević, Milan |
| contents | We deal with the problem of decomposing a complete geometric graph into plane star-forests. In particular, we disprove a recent conjecture by Pach, Saghafian and Schnider by constructing for each $n$ a complete geometric graph on $n$ vertices which can be decomposed into $\frac{n}{2}+1$ plane star-forests. Additionally we prove that for even $n$, every decomposition of complete abstract graph on $n$ vertices into $\frac{n}{2}+1$ star-forests is composed of a perfect matching and $\frac{n}{2}$ star-forests with two edge-balanced components, which we call broken double stars. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11044 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Star-Forest Decompositions of Complete Graphs Antić, Todor Glišić, Jelena Milivojčević, Milan Combinatorics Computational Geometry 05C10 (Primary) 68R10 (Secondary) G.2 We deal with the problem of decomposing a complete geometric graph into plane star-forests. In particular, we disprove a recent conjecture by Pach, Saghafian and Schnider by constructing for each $n$ a complete geometric graph on $n$ vertices which can be decomposed into $\frac{n}{2}+1$ plane star-forests. Additionally we prove that for even $n$, every decomposition of complete abstract graph on $n$ vertices into $\frac{n}{2}+1$ star-forests is composed of a perfect matching and $\frac{n}{2}$ star-forests with two edge-balanced components, which we call broken double stars. |
| title | Star-Forest Decompositions of Complete Graphs |
| topic | Combinatorics Computational Geometry 05C10 (Primary) 68R10 (Secondary) G.2 |
| url | https://arxiv.org/abs/2402.11044 |