41 research outputs found
Counting outerplanar maps
A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our
results, we obtain an e cient scheme for encoding simple outerplanar maps.Peer ReviewedPostprint (published version
Enumeration and random generation of planar maps
This works gives some results on outerplanar maps, which includes a codification of simple outerplanar maps with 3n bits, where n is the number of nodes. It also includes an algorithm to produce random generated maps with equal probability
Unified bijections for planar hypermaps with general cycle-length constraints
We present a general bijective approach to planar hypermaps with two main
results. First we obtain unified bijections for all classes of maps or
hypermaps defined by face-degree constraints and girth constraints. To any such
class we associate bijectively a class of plane trees characterized by local
constraints. This unifies and greatly generalizes several bijections for maps
and hypermaps. Second, we present yet another level of generalization of the
bijective approach by considering classes of maps with non-uniform girth
constraints. More precisely, we consider "well-charged maps", which are maps
with an assignment of "charges" (real numbers) on vertices and faces, with the
constraints that the length of any cycle of the map is at least equal to the
sum of the charges of the vertices and faces enclosed by the cycle. We obtain a
bijection between charged hypermaps and a class of plane trees characterized by
local constraints
Graph limits of random graphs from a subset of connected -trees
For any set of non-negative integers such that and , we consider a random --tree that is uniformly selected from all connected -trees of
vertices where the number of -cliques that contain any fixed -clique
belongs to . We prove that , scaled by
where is the -th Harmonic
number and , converges to the Continuum Random Tree
. Furthermore, we prove the local convergence of the
rooted random --tree to an infinite but
locally finite random --tree .Comment: 21 pages, 6 figure