Large substructures in hypergraphs
Final Report Abstract
This project has contributed to an important field of combinatorics, the search for vertex spanning substructures in large graphs and hypergraphs. There are three main outcomes. Firstly, Sanhueza-Matamala and myself developed a general framework for Hamiltonicity in dense graphs. As applications, we recovered and established so-called Bandwidth Theorems in a variety of settings including Ore-type degree conditions, Pósa-type degree conditions, deficiency-type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders. Secondly, joint work with Alvarado, Kohayakawa, Mota and Stagni has led to a better understanding of the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum degree threshold for loose Hamiltonicity relative to the random hypergraph coincides with its dense analogue. Finally, I have developed an asymptotic characterisation of combinatorial structures that contain perfect tilings. This generalises the geometric theory of hypergraph matching of Keevash and Mycroft. As an application, I recovered recent work on perfect tilings under codegree conditions in hypergraphs, degree conditions in ordered graphs and quasirandom hypergraphs, as well as new bounds for more general degree conditions in hypergraphs.
Publications
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Partitioning a 2-edge-coloured graph of minimum degree 2n/3 + o(n) into 3 monochromatic cycles. European Journal of Combinatorics, 121 (2024, 10), 103838.
Allen, Peter; Böttcher, Julia; Lang, Richard; Skokan, Jozef & Stein, Maya
