Cycles in Random Bipartite Graphs

In this paper we study cycles in random bipartite graph $G(n,n,p)$. We prove that if $p\gg n^{-2/3}$, then $G(n,n,p)$ a.a.s. satisfies the following. Every subgraph $G'\subset G(n,n,p)$ with more than $(1+o(1))n^2p/2$ edges contains a cycle of length $t$ for all even $t\in[4,(1+o(1))n/30]$. Our theorem complements a previous result on bipancyclicity, and is closely related to a recent work of Lee and Samotij.Comment: 8 pages, 2 figure

Tight Hamilton Cycles in Random Uniform Hypergraphs

In this paper we show that $e/n$ is the sharp threshold for the existence of tight Hamilton cycles in random $k$-uniform hypergraphs, for all $k\ge 4$. When $k=3$ we show that $1/n$ is an asymptotic threshold. We also determine thresholds for the existence of other types of Hamilton cycles.Comment: 9 pages. Updated to add materia

Pancyclic Hamilton cycles in random graphs

AbstractLet G(n,p) denote the probability space of the set G of graphs G = (Vn, E) with vertex set Vn = {1,2,…, n} and edges E chosen independently with probability p from E={{u,v}:u,v∈Vn,u≠v}.A graph G∈G(n,p is defined to be pancyclic if, for all s, 3⩽s⩽n there is a cycle of size s on the edges of G. We show that the threshold probability p = (log n + log log n + cn)/n for the property that G contains a Hamilton cycle is also the threshold probability for the existence of a 2-pancyclic Hamilton cycle, which is defined as follows. Given a Hamilton cycle H, we will say that H is k-pancyclic if for each s (3⩽s⩽n−1) we can find a cycle C of length s using only the edges of H and at most k other edges

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