LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Embedding problems in graphs and hypergraphs

In this thesis, we explore several mathematical questions about substructures in graphs and hypergraphs, focusing on algorithmic methods and notions of regularity for graphs and hypergraphs. We investigate conditions for a graph to contain powers of paths and cycles of arbitrary specified linear lengths. Using the well-established graph regularity method, we determine precise minimum degree thresholds for sufficiently large graphs and show that the extremal behaviour is governed by a family of explicitly given extremal graphs. This extends an analogous result of Allen, Böttcher and Hladký for squares of paths and cycles of arbitrary specified linear lengths and confirms a conjecture of theirs. Given positive integers k and j with j < k, we study the length of the longest j-tight path in the binomial random k-uniform hypergraph Hk(n, p). We show that this length undergoes a phase transition from logarithmic to linear and determine the critical threshold for this phase transition. We also prove upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the Pathfinder algorithm, a depth-first search algorithm which discovers j-tight paths in a k-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm finds a long j-tight path. Finally, we investigate the embedding of bounded degree hypergraphs into large sparse hypergraphs. The blow-up lemma is a powerful tool for embedding bounded degree spanning subgraphs with wide-ranging applications in extremal graph theory. We prove a sparse hypergraph analogue of the blow-up lemma, showing that large sparse partite complexes with sufficiently regular small subcomplex counts and no atypical vertices behave as if they were complete for the purpose of embedding complexes with bounded degree and bounded partite structure