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Chordal graphs with bounded tree-width

Chordal graphs with bounded tree-width

Authors :
Castellví, Jordi
Drmota, Michael
Noy, Marc
Requilé, Clément
Publication Year :
2023
Publisher :
arXiv, 2023.

Abstract

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 $c n^{-5/2} \gamma^n n!$, as $n\to\infty$, for some constants $c,\gamma >0$ 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.<br />Comment: 22 pages, 4 figures

Details

Database :
OpenAIRE
Accession number :
edsair.doi.dedup.....f8e52790682f516b5c6b0d3064a68778
Full Text :
https://doi.org/10.48550/arxiv.2301.00194