The evolution of random graphs on surfaces
TL;DR: In this article, the authors investigate the components, subgraphs, maximum degree, and largest face size of S g ( n, m ), finding that there is often different asymptotic behaviour depending on the ratio m n.
About: This article is published in Electronic Notes in Discrete Mathematics.The article was published on 2017-08-01 and is currently open access. It has received 2 citations till now. The article focuses on the topics: Random graph.
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TL;DR: In this paper, the degree of a uniform random vertex in a uniform cograph is of order n, and converges after normalization to the Lebesgue measure on [0, 1].
Abstract: We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence toward a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order n, and converges after normalization to the Lebesgue measure on [0,1]. We finally analyze the vertex connectivity (i.e., the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution of the vertex connectivity is different in the labeled and unlabeled settings. Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.
5 citations
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TL;DR: In this paper, it was shown that the degree of a uniform random vertex in a uniform cograph is of order n and converges after normalization to the Lebesgue measure on 0, 1.
Abstract: We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order $n$, and converges after normalization to the Lebesgue measure on $[0,1]$. We finally analyze the vertex connectivity (i.e. the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution is different in the labeled and unlabeled settings.
Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.
4 citations
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TL;DR: A graph is a planar if it can be embedded in the plane, or in the sphere, so that no two edges cross at an interior point as mentioned in this paper, and a graph together with a particular embedding is called a map.
Abstract: A graph is planar if it can be embedded in the plane, or in the sphere, so that no two edges cross at an interior point A planar graph together with a particular embedding is called a map There is a rich theory of counting maps, started by Tutte in the 1960's However, in this paper we are interested in counting graphs as combinatorial objects, regardless of how many nonequivalent topological embeddings they may have As we are going to see, this makes the counting considerably more difficult
191 citations
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TL;DR: It is shown that the probability that Rn is connected is bounded away from 0 and from 1, and that each positive integer k, with high probability Rn has linearly many vertices of a given degree.
179 citations
Journal Article•
TL;DR: The program plantri as discussed by the authors is the fastest isomorph-free generator of many classes of planar graphs, including triangulations, quadrangulations, and convex polytopes.
Abstract: The program plantri is the fastest isomorph-free generator of many classes of planar graphs, including triangulations, quadrangulations, and convex polytopes. Many applications in the natural sciences as well as in mathematics have appeared. This paper describes plantri's principles of operation, the basis for its efficiency, and the recursive algorithms behind many of its capabilities. In addition, we give many counts of isomorphism classes of planar graphs compiled using plantri. These include triangulations, quadrangulations, convex polytopes, several classes of cubic and quartic graphs, and triangulations of disks.
173 citations
TL;DR: The asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges is derived and it is shown that almost all such graphs with $n$ vertices contain many copies of any fixed planar graph, and this implies that almostAll such graphs have large automorphism groups.
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.
96 citations