Minimum degrees for powers of paths and cycles

We study minimum degree conditions under which a graph $G$ contains $k^{th}$ power of paths and cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of G is large. This extends a result of Allen, B\"ottcher and Hladk\'y concerning the containment of squares of paths and squares of cycles of arbitrary specified lengths and settles a conjecture of theirs in the affirmative.Comment: 69 pages, 3 figures. arXiv admin note: text overlap with arXiv:0906.3299 by other author

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

An approximate blow-up lemma for sparse hypergraphs

We obtain an approximate sparse hypergraph version of the blow-up lemma, showing that partite hypergraphs with sufficient regularity of small subgraph counts behave as if they were complete partite for the purpose of embedding bounded degree hypergraphs

Longest paths in random hypergraphs

The excellent target article of Hamm et al. (2022) raises much food for thought. In this commentary we first discuss what is included in their proposed category of ‘positive evaluations and responses to police assertions of power to attempt social influence’. Given integers k, j with 1 j k − 1, we consider 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 length to linear and determine the critical threshold, as well as proving 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 will find a long j-tight path

Completing graphs to metric spaces

We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions