# Tree-Decorated Planar Maps

@article{Fredes2020TreeDecoratedPM, title={Tree-Decorated Planar Maps}, author={Luis Fredes and Avelio Sep'ulveda}, journal={Electron. J. Comb.}, year={2020}, volume={27}, pages={P1.66} }

We introduce the set of (non-spanning) tree-decorated planar maps, and show that they are in bijection with the Cartesian product between the set of trees and the set of maps with a simple boundary. As a consequence, we count the number of tree decorated triangulations and quadrangulations with a given amount of faces and for a given size of the tree. Finally, we generalise the bijection to study other types of decorated planar maps and obtain explicit counting formulas for them.

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#### References

SHOWING 1-10 OF 40 REFERENCES

Planar Maps as Labeled Mobiles

- Mathematics, Computer Science
- Electron. J. Comb.
- 2004

We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a bijection with a new class… Expand

Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems

- Mathematics, Computer Science
- Electron. J. Comb.
- 2007

It is proved that the bijection presented is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot. Expand

Counting rooted maps by genus II

- Mathematics
- 1972

Abstract Using a combinatorial equivalent for maps, we take the first census of maps on orientable surfaces of arbitrary genus. We generalize to higher genus Tutte's recursion formula for counting… Expand

Bijections for planar maps with boundaries

- Mathematics, Computer Science
- J. Comb. Theory, Ser. A
- 2018

The method is to show that maps with boundaries can be endowed with certain "canonical" orientations, making them amenable to the master bijection approach, developed in previous articles. Expand

Enumeration of almost cubic maps

- Mathematics
- 1972

Abstract This paper deals with the enumeration of rooted planar maps in which the root vertex is of arbitrary valence and all other vertices are trivalent. A formula, in explicit form, is given and… Expand

Counting planar maps, coloured or uncoloured

- Mathematics
- 2011

We present recent results on the enumeration of q-coloured planar maps, where each monochromatic edge carries a weight \nu. This is equivalent to weighting each map by its Tutte polynomial, or to… Expand

A new branch of enumerative graph theory

- Mathematics
- 1962

In a recent survey [ l ] F. Harary pointed out that the problem of enumerating planar graphs was important but still untouched. I am happy to be able to announce some results in this hitherto… Expand

Explicit Enumeration of Triangulations with Multiple Boundaries

- Mathematics, Computer Science
- Electron. J. Comb.
- 2007

R rooted triangulations of a sphere with multiple holes are enumerated by the total number of edges and the length of each boundary component by W.T. Tutte. Expand

A mating-of-trees approach for graph distances in random planar maps

- Mathematics, Physics
- 2017

We introduce a general technique for proving estimates for certain random planar maps which belong to the $$\gamma $$ γ -Liouville quantum gravity (LQG) universality class for $$\gamma \in (0,2)$$ γ… Expand

Uniqueness and universality of the Brownian map

- Mathematics
- 2013

We consider a random planar map Mn which is uniformly distributed over the class of all rooted q-angulations with n faces. We let mn be the vertex set of Mn, which is equipped with the graph distance… Expand