Enumeration of chordal planar graphs and maps
Jordi Castellv\'i, Marc Noy, Cl\'ement Requil\'e

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 vertices, which is asymptotically for a constant and . We also determine the number of rooted simple chordal planar maps with edges, which is asymptotically , where , and 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 by repeatedly adding vertices adjacent to an existing triangular face.
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Taxonomy
TopicsAdvanced Graph Theory Research · Advanced Combinatorial Mathematics · Stochastic processes and statistical mechanics
