Tree-Decorated Planar Maps

  title={Tree-Decorated Planar Maps},
  author={Luis Fredes and Avelio Sep'ulveda},
  journal={Electron. J. Comb.},
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. 
3 Citations
Nonbijective scaling limit of maps via restriction
The main purpose of this work is to provide a framework for proving that, given a family of random maps known to converge in the Gromov–Hausdorff sense, then some (suitable) conditional families ofExpand
External diffusion-limited aggregation on a spanning-tree-weighted random planar map
Let $M$ be the infinite spanning-tree-weighted random planar map, which is the local limit of finite random planar maps sampled with probability proportional to the number of spanning trees theyExpand
Models of random subtrees of a graph
Consider a connected graph $G=(E,V)$ with $N=|V|$ vertices. A subtree of $G$ with size $n$ is a tree which is a subgraph of $G$, with $n$ vertices. When $n=N$, such a subtree is called a spanningExpand


Planar Maps as Labeled Mobiles
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 classExpand
Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems
  • O. Bernardi
  • 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
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 countingExpand
Bijections for planar maps with boundaries
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
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 andExpand
Counting planar maps, coloured or uncoloured
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 toExpand
A new branch of enumerative graph theory
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 hithertoExpand
Explicit Enumeration of Triangulations with Multiple Boundaries
  • M. Krikun
  • 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
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 $$\gamma \in (0,2)$$ γExpand
Uniqueness and universality of the Brownian map
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 distanceExpand