Enumeration of chordal planar graphs and maps

TL;DR

This paper provides asymptotic formulas for counting labelled chordal planar graphs and maps, revealing their growth rates and structural properties through combinatorial and singularity analysis.

Contribution

It introduces exact asymptotic enumeration formulas for chordal planar graphs and maps, expanding understanding of their combinatorial complexity.

Findings

Number of labelled chordal planar graphs grows as c_1·n^{-5/2}·γ^n·n! with γ≈11.89235

Number of rooted simple chordal planar maps grows as c_2·n^{-3/2}·δ^n with δ≈6.40375

Chordal planar graphs include rich 3-connected members like chordal triangulations.

Abstract

We determine the number of labelled chordal planar graphs with $n$ vertices, which is asymptotically $c_1\cdot n^{-5/2} \gamma^n n!$ for a constant $c_1>0$ and $\gamma \approx 11.89235$. We also determine the number of rooted simple chordal planar maps with $n$ edges, which is asymptotically $c_2 n^{-3/2} \delta^n$, where $\delta = 1/\sigma \approx 6.40375$, and $\sigma$ is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively rich. The 3-connected members are precisely chordal triangulations, those obtained starting from $K_4$ by repeatedly adding vertices adjacent to an existing triangular face.