Descripción del proyecto
Un nuevo estudio se propone resolver problemas relacionados con los subgrafos generadores
La teoría de grafos, el estudio de estructuras matemáticas empleadas para elaborar modelos de relaciones entre pares de objetos, presenta numerosas conexiones con otros ámbitos, como la combinatoria y la informática teórica. Un metaproblema fundamental de la teoría de grafos es el siguiente: dado un grafo H, ¿qué condiciones garantizan que otro grafo G contenga una copia de H como subgrafo? El proyecto financiado con fondos europeos SSiGraph abordará una serie de problemas extremales y probabilísticos difíciles e interesantes sobre subgrafos generadores en grafos aleatorios y coloridos. Los objetivos concretos representan una serie cuidadosamente seleccionada de problemas conexos destacados, cuya solución podría marcar un progreso verdaderamente importante en este campo.
Objetivo
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.
Ámbito científico
Programa(s)
Régimen de financiación
ERC-STG - Starting GrantInstitución de acogida
CV4 8UW COVENTRY
Reino Unido