Chordal graphs with bounded tree-width

Abstract

Given $t\ge 2$ and $0\le k\le t$, 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 ($2\le i\le t$) 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 $t\ge 3$. 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 (1985) [21], were an algorithm is developed to obtain the exact number of labelled chordal graphs on n vertices.