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

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

Generating random graphs in biased Maker-Breaker games

We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for $b=o\left(\sqrt{n}\right)$, Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a $(1:b)$ game on $E(K_n)$. As another application, we show that for $b=\Theta\left(n/\ln n\right)$, playing a $(1:b)$ game on $E(K_n)$, Maker can build a graph which contains copies of all spanning trees having maximum degree $\Delta=O(1)$ with a bare path of linear length (a bare path in a tree $T$ is a path with all interior vertices of degree exactly two in $T$)

Pancyclicity of Hamiltonian and highly connected graphs

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $\alpha(G)$ for the independence number of $G$, i.e. the size of the largest subset of the vertex set that does not contain an edge, and $\kappa(G)$ for the (vertex) connectivity, i.e. the size of the smallest subset of the vertex set that can be deleted to obtain a disconnected graph. A celebrated theorem of Chv\'atal and Erd\H{o}s says that $G$ is Hamiltonian if $\kappa(G) \ge \alpha(G)$. Moreover, Bondy suggested that almost any non-trivial conditions for Hamiltonicity of a graph should also imply pancyclicity. Motivated by this, we prove that if $\kappa(G) \ge 600\alpha(G)$ then G is pancyclic. This establishes a conjecture of Jackson and Ordaz up to a constant factor. Moreover, we obtain the more general result that if G is Hamiltonian with minimum degree $\delta(G) \ge 600\alpha(G)$ then G is pancyclic. Improving an old result of Erd\H{o}s, we also show that G is pancyclic if it is Hamiltonian and $n \ge 150\alpha(G)^3$. Our arguments use the following theorem of independent interest on cycle lengths in graphs: if $\delta(G) \ge 300\alpha(G)$ then G contains a cycle of length $\ell$ for all $3 \le \ell \le \delta(G)/81$.Comment: 15 pages, 1 figur

Fast Strategies in Waiter-Client Games on KnK_n

Waiter-Client games are played on some hypergraph $(X,\mathcal{F})$, where $\mathcal{F}$ denotes the family of winning sets. For some bias $b$, during each round of such a game Waiter offers to Client $b+1$ elements of $X$, of which Client claims one for himself while the rest go to Waiter. Proceeding like this Waiter wins the game if she forces Client to claim all the elements of any winning set from $\mathcal{F}$. In this paper we study fast strategies for several Waiter-Client games played on the edge set of the complete graph, i.e. $X=E(K_n)$, in which the winning sets are perfect matchings, Hamilton cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.Comment: 38 page