On enumeration of tree-rooted planar cubic maps

TL;DR

This paper counts specific planar cubic maps with marked structures, revealing a precise enumeration formula involving Catalan numbers for objects with 2n vertices.

Contribution

It provides an exact enumeration formula for planar cubic maps with a marked spanning tree and directed edge, linking combinatorial structures to Catalan numbers.

Findings

Number of such maps with 2n vertices is C_{2n} * C_{n+1}

Establishes a connection between planar cubic maps and Catalan numbers

Offers a combinatorial enumeration result for specific graph embeddings

Abstract

We consider planar cubic maps, i.e. connected cubic graphs imbedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number.