Spanning Subgraphs in Graphs

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.