The Degree Sequence of Random Graphs from Subcritical Classes

Abstract

In this work we determine the expected number of vertices of degree k = k(n) in a graph with n vertices that is drawn uniformly at random from a subcritical graph class. Examples of such classes are outerplanar, series-parallel, cactus and clique graphs. Moreover, we provide exponentially small bounds for the probability that the quantities in question deviate from their expected values. † Parts of this work appeared as an extended abstract in N. Bernasconi, K. Panagiotou and A. Steger, 'On the degree sequences of random outerplanar and series-parallel graphs ', in APPROX-RANDOM 2008 , pp. 303-316. at https:/www.cambridge.org/core/terms. https://doi.The picture changes dramatically if we are interested in natural graph classes. A standard example that has evolved over the last decade as a reference model in this context is the class of planar graphs. The random planar graph R n was first investigated in [7] by Denise, Vasconcellos and Welsh and has attracted considerable attention since then. We mention selectively a few results. McDiarmid, Steger and Welsh [16] showed the surprising fact that R n does not share the 0-1 law known from standard random graph theory: the probability of connectedness is bounded away from 0 and 1 by positive constant values; moreover, the situation is similar if the average degree is fixed [13]. These results relied on a (crude) counting of the number of planar graphs with n vertices. A breakthrough occurred with the recent results of Giménez and Noy [14], who not only managed to determine the asymptotic value of the number of planar graphs with n vertices, but also showed that the number of edges in R n is asymptotically normally distributed. Moreover, they studied the number of connected and 2-connected components in R n . The proofs of these results are based on singularity analysis of generating functions, a powerful method from analytic combinatorics that has led to many beautiful results: see the book by Flajolet and Sedgewick [10].Our results. In this paper we further elaborate and extend significantly an approach that was used in [3] to obtain the degree sequence and subgraph counts of random dissections of convex polygons. More precisely, we exploit the so-called Boltzmann sampler framework by Duchon, Flajolet, Louchard and Schaeffer [9] to reduce the study of degree sequences to properties of sequences of independent and identically distributed random variables. Hence, we can -and do -use many tools developed in classical random graph theory to obtain extremely tight results.Our first main contribution is a general framework that allows us to derive mechanically the degree distribution of random graphs from certain 'nice' graph classes, which are 'subcritical' in a well-defined analytic sense: see Section 3 for details. Our framework can be readily applied to obtain the degree sequence of random graphs from 'simple' classes, such as Cayley treesi.e., (non-plane) labelled trees -or graphs which have the property that their maximal 2-connected components (or equivalently,...

The maximum degree of random planar graphs

“…Such a result has been proved recently using complex analytic methods by Drmota, Giménez and Noy [12] (with a different value of w 0 ) for series-parallel graphs, an important subclass of planar graphs. The result for series-parallel graphs was previously conjectured by Bernasconi, Panagiotou and Steger [2], where the authors prove strong concentration results for the number of vertices of degree up to (c − ) log n, where c = 1/ log w 0 . The results in [2] are obtained using so-called Boltzmann samplers, a framework that reduces the study of vertex degrees to properties of sequences of independent and identically distributed (i.i.d.)…”

Extremal Parameters in Sub-Critical Graph Classes

“…Several well known graph classes are sub-critial. For expample, labelled trees, cacti-graphs, outerplanar graphs and series-parallel graphs are sub-critical, see [4,7]. However, the class of labelled planar graphs is not sub-critical.…”

“…Sub-critical graph classes include important graph classes like trees, outerplanar graphs, or series-parallel graphs. During the last few years these kind of graphs have been studied from various points of view [4,7,8,9]. In particular we mention here the paper by Drmota et al [7], where it has been shown that several (additive) parameters like the number of edges, the number of blocks or the number of vertices of given degree satisfy a central limit theorem.…”

Maximum degree in minor-closed classes of graphs

“…Analogous results have been proved for series-parallel and outerplanar graphs [4], with suitable constants. Using the framework of Boltzmann samplers, results about the degree distribution of subcritical graph classes such as outerplanar graphs, series-parallel graphs, cactus graphs and clique graphs can also be found in [1]. This paper also contains conjectures of the exact values of c OP (c SP , respectively) so that the maximum degree in outerplanar graphs (series-parallel graphs, respectively) will be roughly c OP log n (c SP log n, respectively).…”