HONG LIU   刘鸿

Embeddings in Sparse Graphs

Welcome to the webpage of the UKRI Future Leaders Fellowship Project "Embeddings in Sparse Graphs" (EiSG), 1 May 2019 - 30 April 2023.

Summary

Extremal combinatorics has witnessed a rapid growth in the past few decades. An archetype problem in this field is to study the structure of subgraphs appearing in different classes of graphs. Such problems are much better understand when the host graphs are dense thanks to a variety of tools applicable to dense graphs. This project will focus on such problems for sparse host graphs. In practice, almost all the graphs used to model real-life networks, such as facebook graphs, (artificial) neural networks, are sparse.

The goal is to develop a theory of certain sublinear expanders that can be used to construct subgraphs with various properties in graphs with (sufficiently large) constant average degree. This project will connect sublinear expanders with two other central notions in combinatorics: cycles and topological minors. The main objective is to design and construct versatile building blocks in sparse graphs with the aid of expansion properties. Such building blocks will then be used to embed subgraphs with various complex structure. The theory developed will make decisive progress on central problems concerning embedding sparse subgraphs with additional arithmetic properties. Some of such problems have remained wide open since the 60s despite active attempts from various researchers.

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