Chordal graphs with bounded tree-width
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916111446966272 |
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| author | Castellví, Jordi Drmota, Michael Noy, Marc Requilé, Clément |
| author_facet | Castellví, Jordi Drmota, Michael Noy, Marc Requilé, Clément |
| contents | Given $t\geq 2$ and $0\leq k\leq t$, we prove that the number of labelled $k$-connected chordal graphs with $n$ vertices and tree-width at most $t$ is asymptotically $c n^{-5/2} γ^n n!$, as $n\to\infty$, for some constants $c,γ>0$ depending on $t$ and $k$. Additionally, we show that the number of $i$-cliques ($2\leq i\leq t$) in a uniform random $k$-connected chordal graph with tree-width at most $t$ is normally distributed as $n\to\infty$.
The asymptotic enumeration of graphs of tree-width at most $t$ is wide open for $t\geq 3$. To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald [Counting Labelled Chordal Graphs, Graphs and Combinatorics (1985)], were an algorithm is developed to obtain the exact number of labelled chordal graphs on $n$ vertices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_00194 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Chordal graphs with bounded tree-width Castellví, Jordi Drmota, Michael Noy, Marc Requilé, Clément Combinatorics 05C30 Given $t\geq 2$ and $0\leq k\leq t$, we prove that the number of labelled $k$-connected chordal graphs with $n$ vertices and tree-width at most $t$ is asymptotically $c n^{-5/2} γ^n n!$, as $n\to\infty$, for some constants $c,γ>0$ depending on $t$ and $k$. Additionally, we show that the number of $i$-cliques ($2\leq i\leq t$) in a uniform random $k$-connected chordal graph with tree-width at most $t$ is normally distributed as $n\to\infty$. The asymptotic enumeration of graphs of tree-width at most $t$ is wide open for $t\geq 3$. To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald [Counting Labelled Chordal Graphs, Graphs and Combinatorics (1985)], were an algorithm is developed to obtain the exact number of labelled chordal graphs on $n$ vertices. |
| title | Chordal graphs with bounded tree-width |
| topic | Combinatorics 05C30 |
| url | https://arxiv.org/abs/2301.00194 |