Projektbeschreibung
Neue Untersuchung soll Probleme im Zusammenhang mit aufspannenden Teilgraphen lösen
Die Graphentheorie – das Studium mathematischer Strukturen, die zur Modellierung paarweiser Beziehungen zwischen Objekten verwendet werden, – weist zahlreiche natürliche Verbindungen zu anderen Fachgebieten wie etwa der Kombinatorik und der theoretischen Informatik auf. Ein fundamentales Metaproblem der Graphentheorie ist folgendes: Welche Bedingungen garantieren bei einem gegebenen Graphen H, dass ein anderer Graph G eine Kopie von H als Teilgraph enthält? Das EU-finanzierte Projekt SSiGraph wird sich mit einer Reihe spannender und herausfordernder extremaler und probabilistischer Probleme der aufspannenden Teilgraphen in zufälligen und farbigen Graphen befassen. Die spezifischen Ziele repräsentieren eine sorgfältige Auswahl an zusammenhängenden großen offenen Problemen, deren Lösung als wahrhaft bedeutender Fortschritt auf diesem Gebiet gelten dürfte.
Ziel
Graph Theory is a highly active area of Combinatorics with strong links to fields such as Optimisation and Theoretical Computer Science. A fundamental meta-problem in Graph Theory is the following: given a graph H, what conditions guarantee that another graph G contains a copy of H as a subgraph? This is particularly important when H is spanning, i.e. where G and H have the same number of vertices.
This project will address a range of exciting and challenging extremal and probabilistic problems on spanning subgraphs in graphs, in the following two interrelated areas:
1. Spanning subgraphs in random graphs: A key aim of Probabilistic Combinatorics is to determine the density threshold for the appearance of different subgraphs in random graphs. This is particularly difficult when the subgraph is spanning, where the known results and techniques are typically highly specific. This project will lead to a unified paradigm for studying thresholds of spanning subgraphs by introducing and developing a new coupling technique. This will provide an excellent platform to study the Kahn-Kalai conjecture, a bold general conjecture on appearance thresholds, and problems including hitting-time conjectures and universality problems.
2. Spanning subgraphs in coloured graphs: Many different combinatorial problems are expressible using edge coloured graphs, including Latin square problems dating back to Euler. My objectives here concern long-standing problems on spanning trees, cycles and matchings, and, through this, the resolution of several famous labelling and packing problems.
In preliminary work I have developed techniques to study these problems, techniques which will have a far reaching impact, and certainly lead to further applications, e.g. with hypergraphs and resilience problems. The objectives represent a carefully selected range of related major outstanding problems, whose solution would mark truly significant progress in the field.
Wissenschaftliches Gebiet
Programm/Programme
Thema/Themen
Finanzierungsplan
ERC-STG - Starting GrantGastgebende Einrichtung
CV4 8UW COVENTRY
Vereinigtes Königreich