Results 1  10
of
20
Models of Random Regular Graphs
 In Surveys in combinatorics
, 1999
"... In a previous paper we showed that a random 4regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random dregular g ..."
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Cited by 157 (33 self)
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In a previous paper we showed that a random 4regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random dregular graph for other d between 5 and 10 inclusive is a.a.s. restricted to a range of two integer values: {3, 4} for d = 5, {4, 5} for d = 7, 8, 9, and {5, 6} for d = 10. The proof uses efficient algorithms which a.a.s. colour these random graphs using the number of colours specified by the upper bound. These algorithms are analysed using the differential equation method, including an analysis of certain systems of differential equations with discontinuous right hand sides. 1
Random planar lattices and integrated superBrownian excursion
 Probab. Th. Rel. Fields
"... Abstract. In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scalin ..."
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Cited by 63 (3 self)
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Abstract. In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r = R−L of the support of the onedimensional ISE, or precisely: n −1/4 rn law − → (8/9) 1/4 r. More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive halfline, reminiscent of Cori and Vauquelin’s well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat’s construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks (e (n) , ˆ W (n) ) to the Brownian snake description (e, ˆ W) of ISE. Our results suggest the existence of a Continuum Random Map describing in term of ISE the scaled limit of the dynamical triangulations considered in twodimensional pure quantum gravity. 1.
Random maps, coalescing saddles, singularity analysis, and Airy phenomena
 Random Structures & Algorithms
, 2001
"... A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the ..."
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Cited by 46 (6 self)
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.
Optimal Coding and Sampling of Triangulations
, 2003
"... Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a bypr ..."
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Cited by 39 (5 self)
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Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1
The number of labeled 2connected planar graphs
 Journal of Combinatorics
, 2000
"... We derive the asymptotic expression for the number of labeled 2connected 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 gro ..."
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Cited by 36 (2 self)
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We derive the asymptotic expression for the number of labeled 2connected 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.
Growth and percolation on the uniform infinite planar triangulation
 Geom. Funct. Anal
, 2003
"... A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in [4], is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT. By analyzing the progress rate of the growth process we show that a. ..."
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Cited by 27 (2 self)
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A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in [4], is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT. By analyzing the progress rate of the growth process we show that a.s. the UIPT has growth rate r 4 up to polylogarithmic factors, confirming heuristic results from the physics literature. Additionally, the boundary component of the ball of radius r separating it from infinity a.s. has growth rate r 2 up to polylogarithmic factors. It is also shown that the properly scaled size of a variant of the free triangulation of an mgon (also defined in [4]) converges in distribution to an asymmetric stable random variable of type 1/2. By combining Bernoulli site percolation with the growth process for the UIPT, it is shown that a.s. the critical probability pc = 1/2 and that at pc percolation does not occur. Subject classification: Primary 05C80; Secondary 05C30, 82B43, 81T40. 1
Limits of normalized quadrangulations. The Brownian map
 Ann. Probab
, 2004
"... Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper, we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name t ..."
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Cited by 20 (0 self)
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Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper, we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations and the model of Brownian map is defined. The weak convergences hold in these metric spaces. 1
The Distribution of the Maximum Vertex Degree in Random Planar Maps
 J. Combin. Theory Ser. A
, 2000
"... We determine the limiting distribution of the maximum vertex degree n in a random triangulation of an ngon, 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. ..."
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Cited by 18 (2 self)
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We determine the limiting distribution of the maximum vertex degree n in a random triangulation of an ngon, 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...
The Size Of The Largest Components In Random Planar Maps
, 1999
"... . Bender, Richmond, and Wormald showed that in almost all planar 3connected triangulations (or dually, 3connected cubic maps) with n edges, the largest 4connected triangulation (or dually, the largest cyclically 4edgeconnected cubic component) has about
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Cited by 17 (4 self)
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.<F3.837e+05> Bender, Richmond, and Wormald showed that in almost all planar 3connected triangulations (or dually, 3connected cubic maps) with<F3.083e+05> n<F3.837e+05> edges, the largest 4connected triangulation (or dually, the largest cyclically 4edgeconnected cubic component) has about<F3.083e+05><F3.837e+05> n/2 edges<F3.086e+05> [Random Structures Algorithms,<F3.837e+05> 7 (1995), pp. 273285]. In this paper, we derive some general results about the size of the largest component and apply them to a variety of types of planar maps.<F4.005e+05> Key words.<F3.837e+05> planar map, 4connected component, triangulation, cubic graph<F4.005e+05> AMS subject classifications.<F3.837e+05> 05C30, 05C40<F4.005e+05> PII.<F3.837e+05> S0895480195292053<F4.835e+05> 1. Introduction.<F4.426e+05> Recently, Bender, Richmond, and Wormald [3] showed that in almost all 3connected triangulations with<F4.309e+05> n<F4.426e+05> edges, the largest 4connected triangulation has<F4.309e+05><F4.426e+05>...