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

Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights

Let $G$ be a graph and $\mathcal {S}$ be a subset of $Z$. A vertex-coloring $\mathcal {S}$-edge-weighting of $G$ is an assignment of weight $s$ by the elements of $\mathcal {S}$ to each edge of $G$ so that adjacent vertices have different sums of incident edges weights. It was proved that every 3-connected bipartite graph admits a vertex-coloring $\{1,2\}$-edge-weighting (Lu, Yu and Zhang, (2011) \cite{LYZ}). In this paper, we show that the following result: if a 3-edge-connected bipartite graph $G$ with minimum degree $\delta$ contains a vertex $u\in V(G)$ such that $d_G(u)=\delta$ and $G-u$ is connected, then $G$ admits a vertex-coloring $\mathcal {S}$-edge-weighting for $\mathcal {S}\in \{\{0,1\},\{1,2\}\}$. In particular, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring $\mathcal {S}$-edge-weighting for $\mathcal {S}\in \{\{0,1\},\{1,2\}\}$. The bound is sharp, since there exists a family of infinite bipartite graphs which are 2-connected and do not admit vertex-coloring $\{1,2\}$-edge-weightings or vertex-coloring $\{0,1\}$-edge-weightings.Comment: In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring S-edge-weighting for S\in {{0,1},{1,2}

Regular spanning subgraphs of bipartite graphs of high minimum degree

Let G be a simple balanced bipartite graph on $2n$ vertices, $\delta = \delta(G)/n$, and $\rho={\delta + \sqrt{2 \delta -1} \over 2}$. If $\delta > 1/2$ then it has a $\lfloor \rho n \rfloor$-regular spanning subgraph. The statement is nearly tight.Comment: submitte