### Description

Project Summary “Asymptotic properties of graphs on a surface”

Since the foundation of the theory of random graphs by Erdős and Rényi five decades ago, various random graphs have been introduced and studied. One example is random graphs on a surface, in particular random planar graphs. Graphs on a 2-dimensional surface and related objects (e.g. planar graphs, triangulations) have been among the most studied objects in graph theory, enumerative combinatorics, discrete probability theory, and statistical physics.

The main objectives of this project are to study the asymptotic properties and limit behaviour of random graphs on a surface (e.g. evolution, phase transition, critical behaviour, component size distribution) and to investigate enumerative and algorithmic aspects of unlabelled graphs on a surface (e.g. connectivity, symmetry, decomposition, random generation). This project aims to provide solutions to important and challenging open problems that call for further focused effort in view of recent developments in the field. The subject of the project is structured into three guiding themes, which are closely related and in which the following goals will be accomplished.

I. Random graphs on a surface

• Component structure of random complex planar graphs

• Critical behaviour of random graphs on a surface

• Threshold for the chromatic number of random planar graphs

II. Enumeration of unlabelled graphs on a surface

• Asymptotic number of unlabelled planar graphs

• Asymptotic number of unlabelled graphs on a surface with positive genus

• Systematic strategy for a constructive decomposition along symmetries

III. Walsh-Boltzmann sampler for unlabelled graphs on a surface

• Walsh-Boltzmann sampler for planar case

• Walsh-Boltzmann sampler for arbitrary genus case

• Decomposition strategy incorporating symmetries and cycle-pointing

In comparison with the classical Erdős-Rényi random graph, additional constraints imposed on graphs (e.g. planarity, genus, symmetry) lead to serious difficulties in the analysis and require interplay between the theory of random graphs, structural graph theory, enumerative combinatorics, analytic combinatorics, and algorithmic graph theory. To achieve the objectives of the project, it will be necessary to advance existing approaches and develop new techniques which combine probabilistic methods, graph theoretic methods, methods from analytic combinatorics (e.g. singularity analysis, saddle point method), and algorithmic methods (e.g. Boltzmann sampler).

Since the foundation of the theory of random graphs by Erdős and Rényi five decades ago, various random graphs have been introduced and studied. One example is random graphs on a surface, in particular random planar graphs. Graphs on a 2-dimensional surface and related objects (e.g. planar graphs, triangulations) have been among the most studied objects in graph theory, enumerative combinatorics, discrete probability theory, and statistical physics.

The main objectives of this project are to study the asymptotic properties and limit behaviour of random graphs on a surface (e.g. evolution, phase transition, critical behaviour, component size distribution) and to investigate enumerative and algorithmic aspects of unlabelled graphs on a surface (e.g. connectivity, symmetry, decomposition, random generation). This project aims to provide solutions to important and challenging open problems that call for further focused effort in view of recent developments in the field. The subject of the project is structured into three guiding themes, which are closely related and in which the following goals will be accomplished.

I. Random graphs on a surface

• Component structure of random complex planar graphs

• Critical behaviour of random graphs on a surface

• Threshold for the chromatic number of random planar graphs

II. Enumeration of unlabelled graphs on a surface

• Asymptotic number of unlabelled planar graphs

• Asymptotic number of unlabelled graphs on a surface with positive genus

• Systematic strategy for a constructive decomposition along symmetries

III. Walsh-Boltzmann sampler for unlabelled graphs on a surface

• Walsh-Boltzmann sampler for planar case

• Walsh-Boltzmann sampler for arbitrary genus case

• Decomposition strategy incorporating symmetries and cycle-pointing

In comparison with the classical Erdős-Rényi random graph, additional constraints imposed on graphs (e.g. planarity, genus, symmetry) lead to serious difficulties in the analysis and require interplay between the theory of random graphs, structural graph theory, enumerative combinatorics, analytic combinatorics, and algorithmic graph theory. To achieve the objectives of the project, it will be necessary to advance existing approaches and develop new techniques which combine probabilistic methods, graph theoretic methods, methods from analytic combinatorics (e.g. singularity analysis, saddle point method), and algorithmic methods (e.g. Boltzmann sampler).

Status | Finished |
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Effective start/end date | 1/03/15 → 31/03/19 |