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Electr. J. Comb. 01/2010; 17.
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Random Struct. Algorithms. 01/2008; 32:38-48.
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ABSTRACT: We give a general result showing that the asymptotic behaviour of high moments determines the shape of distributions which are asymptotically normal. Both the factorial and non-factorial (non-central) moments are treated. This differs from the usual moment method in combinatorics, as the expected value may tend to infinity quite rapidly. Applications are given to submap counts in random planar triangulations, where we use a simple argument to asymptotically determine high moments for the number of copies of a given subtriangulation in a random 3-connected planar triangulation. Similar results are also obtained for 2-connected triangulations and quadrangulations with no multiple edges.
Probability Theory and Related Fields 10/2004; 130(3):368-376. · 1.53 Impact Factor
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Combinatorica. 01/2003; 23:467-486.
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ABSTRACT: We derive the asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many copies of any fixed planar graph, and this implies that almost all such graphs have large automorphism groups.
12/2002;
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ABSTRACT: Although much work has been done on enumerating rooted planar maps since Tutte's poineering works in early 1960s, many classes of maps with no loops or multiple edges are still untreated. In this paper, we enumerate three classes of cubic planar maps with no loops or multiple edges: 1-connected; 2-connected; 3-connected and triangle-free. Research supported by NSERCC, the University of Melbourne and the Australian Research Council y Research supported by the Australian Research Council 1 1 Introduction We consider planar maps in this paper. These are connected graphs embedded in the plane. A planar map is rooted if an edge is distinguished together with a vertex on the edge and a side of the edge. The face on the distinguished side is called the root face. For planar maps, by convention the root face must be the unbounded face. (Thus a rooted planar map is equivalent to a rooted map on the sphere with arbitrary root face.) Much work has been done on enumerating rooted planar ma...
02/2000;
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ABSTRACT: We determine the limiting distribution of the maximum vertex degree n in a random triangulation of an n-gon, and show that it is the same as that of the maximum of n independent identically distributed random variables G 2 , where G 2 is the sum of two independent geometric(1=2) random variables. this answers armatively a question of Devroye, Flajolet, Hurtado, Noy and Steiger, who gave much weaker almost sure bounds on n . An interesting consequence of this is that the asymptotic probability that a random triangulation has a unique vertex with maximum degree is about 0:72. We also give an analogous result for random planar maps in general. Research supported by NSERCC y Research supported by the Australian Research Council 1 1 Introduction Throughout this paper, a map is a connected graph G embedded in the plane with no edge crossings. Loops and multiple edges are allowed in G. A map is rooted if an edge is distinguished together with a vertex on the edge and a side of th...
12/1999;
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SIAM J. Discrete Math. 01/1999; 12:217-228.
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ABSTRACT: In this paper we obtain asymptotics for the number of rooted 3-connected maps on an arbitrary surface and use them to prove that almost all rooted 3-connected maps on any fixed surface have large edge-width and large face-width. It then follows from the result of Roberston and Vitray [10] that almost all rooted 3-connected maps on any fixed surface are minimum genus embeddings and their underlying graphs are uniquely embeddable on the surface.
Journal of the Australian Mathematical Society 01/1996; 60(01):31 - 41. · 0.42 Impact Factor
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Random Struct. Algorithms. 01/1995; 7:273-286.
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ABSTRACT: We show that every triangulation of a disk or an annulus has a spanning Eulerian subgraph with maximum degree eight.
Since every triangulation in the projective plane, the torus and the Klein bottle has a spanning subgraph which triangulates
an annulus, this implies that all triangulations in the projective plane, the torus and the Klein bottle have spanning Eulerian
subgraphs with maximum degree at most eight.
Graphs and Combinatorics 05/1994; 10(2):123-131. · 0.32 Impact Factor
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Acta Inf. 01/1987; 24:475-489.
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Combinatorica. 01/1986; 6:15-22.
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Discrete Mathematics. 01/1985; 54:235-237.
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ABSTRACT: We obtain asymptotics for the number of rooted nonseparable maps on an arbitrary surface. A nonsingular map is defined to be a map with no multiple vertex-face incidences. Trivially, every nonsingular map is nonseparable. We show that almost all nonseparable maps on a given surface are nonsingular.
Journal of Combinatorial Theory, Series A.
