Long paths and cycles in random subgraphs of graphs with large minimum degree

For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge \omega/k$ for a function $\omega=\omega(k)$ that tends to infinity as $k$ does, then $G_p$ asymptotically almost surely contains a cycle (and thus a path) of length at least $(1-o(1))k$, and (ii) if $p \ge (1+o(1))\ln k/k$, then $G_p$ asymptotically almost surely contains a path of length at least $k$. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking $G$ to be the complete graph on $k+1$ vertices.Comment: 26 page

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

Cycle packing

In the 1960s, Erd\H{o}s and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(n log n) upper bound by repeatedly removing the edges of the longest cycle. We make the first progress on this problem, showing that O(n log log n) cycles and edges suffice. We also prove the Erd\H{o}s-Gallai conjecture for random graphs and for graphs with linear minimum degree.Comment: 18 page

On random k-out sub-graphs of large graphs

We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in all. When the minimum degree $\delta(G)\geq (\frac12+\epsilon)n,\,n=|V|$ then $G_k$ is $k$-connected w.h.p. for $k=O(1)$; Hamiltonian for $k$ sufficiently large. When $\delta(G) \geq m$, then $G_k$ has a cycle of length $(1-\epsilon)m$ for $k\geq k_\epsilon$. By w.h.p. we mean that the probability of non-occurrence can be bounded by a function $\phi(n)$ (or $\phi(m)$) where $\lim_{n\to\infty}\phi(n)=0$

Hamilton decompositions of regular tournaments

We show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each \eta>0 every regular tournament G of sufficiently large order n contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968. Our result also extends to almost regular tournaments.Comment: 38 pages, 2 figures. Added section sketching how we can extend our main result. To appear in the Proceedings of the LM

Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved