1,622 research outputs found

    Embedding large subgraphs into dense graphs

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    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

    On globally sparse Ramsey graphs

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    We say that a graph GG has the Ramsey property w.r.t.\ some graph FF and some integer r2r\geq 2, or GG is (F,r)(F,r)-Ramsey for short, if any rr-coloring of the edges of GG contains a monochromatic copy of FF. R{\"o}dl and Ruci{\'n}ski asked how globally sparse (F,r)(F,r)-Ramsey graphs GG can possibly be, where the density of GG is measured by the subgraph HGH\subseteq G with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs FF. In this work we determine the Ramsey density up to some small error terms for several cases when FF is a complete bipartite graph, a cycle or a path, and r2r\geq 2 colors are available

    On-line Ramsey numbers

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    Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colours them, in either red or blue, as each appears. Builder's aim is to force Painter to draw a monochromatic copy of a fixed graph G. The minimum number of edges which Builder must draw, regardless of Painter's strategy, in order to guarantee that this happens is known as the on-line Ramsey number \tilde{r}(G) of G. Our main result, relating to the conjecture that \tilde{r}(K_t) = o(\binom{r(t)}{2}), is that there exists a constant c > 1 such that \tilde{r}(K_t) \leq c^{-t} \binom{r(t)}{2} for infinitely many values of t. We also prove a more specific upper bound for this number, showing that there exists a constant c such that \tilde{r}(K_t) \leq t^{-c \frac{\log t}{\log \log t}} 4^t. Finally, we prove a new upper bound for the on-line Ramsey number of the complete bipartite graph K_{t,t}.Comment: 11 page

    Density theorems for bipartite graphs and related Ramsey-type results

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    In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements
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