Decomposing random regular graphs into stars
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915797102755840 |
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| author | Delcourt, Michelle Greenhill, Catherine Isaev, Mikhail Lidický, Bernard Postle, Luke |
| author_facet | Delcourt, Michelle Greenhill, Catherine Isaev, Mikhail Lidický, Bernard Postle, Luke |
| contents | We study $k$-star decompositions, that is, partitions of the edge set into disjoint stars with $k$ edges, in the uniformly random $d$-regular graph model $\mathcal{G}_{n,d}$. Using the small subgraph conditioning method, we prove an existence result for such decompositions for all $d,k$ such that $d/2 < k \leq d/2 + \max\{1,\frac{1}{6}\log d\}$. More generally, we give a sufficient existence condition that can be checked numerically for any given values of $d$ and $k$. Complementary negative results are obtained using the independence ratio of random regular graphs. Our results establish an existence threshold for $k$-star decompositions in $\mathcal{G}_{n,d}$ for all $d\leq 100$ and $k > d/2$.
For smaller values of $k$, the connection between $k$-star decompositions and $β$-orientations allows us to apply results of Thomassen (2012) and Lovász, Thomassen, Wu and Zhang (2013). We prove that random $d$-regular graphs satisfy their assumptions with high probability, thus establishing a.a.s. existence of $k$-star decompositions (i) when $2k^2+k\leq d$, and (ii) when $k$ is odd and $k < d/2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_16037 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Decomposing random regular graphs into stars Delcourt, Michelle Greenhill, Catherine Isaev, Mikhail Lidický, Bernard Postle, Luke Combinatorics We study $k$-star decompositions, that is, partitions of the edge set into disjoint stars with $k$ edges, in the uniformly random $d$-regular graph model $\mathcal{G}_{n,d}$. Using the small subgraph conditioning method, we prove an existence result for such decompositions for all $d,k$ such that $d/2 < k \leq d/2 + \max\{1,\frac{1}{6}\log d\}$. More generally, we give a sufficient existence condition that can be checked numerically for any given values of $d$ and $k$. Complementary negative results are obtained using the independence ratio of random regular graphs. Our results establish an existence threshold for $k$-star decompositions in $\mathcal{G}_{n,d}$ for all $d\leq 100$ and $k > d/2$. For smaller values of $k$, the connection between $k$-star decompositions and $β$-orientations allows us to apply results of Thomassen (2012) and Lovász, Thomassen, Wu and Zhang (2013). We prove that random $d$-regular graphs satisfy their assumptions with high probability, thus establishing a.a.s. existence of $k$-star decompositions (i) when $2k^2+k\leq d$, and (ii) when $k$ is odd and $k < d/2$. |
| title | Decomposing random regular graphs into stars |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2308.16037 |