ﻻ يوجد ملخص باللغة العربية
Kim, Kuhn, Osthus and Tyomkyn (Trans. Amer. Math. Soc. 371 (2019), 4655--4742) greatly extended the well-known blow-up lemma of Komlos, Sarkozy and Szemeredi by proving a `blow-up lemma for approximate decompositions which states that multipartite quasirandom graphs can be almost decomposed into any collection of bounded degree graphs with the same multipartite structure and slightly fewer edges. This result has already been used by Joos, Kim, Kuhn and Osthus to prove the tree packing conjecture due to Gyarfas and Lehel from 1976 and Ringels conjecture from 1963 for bounded degree trees as well as implicitly in the recent resolution of the Oberwolfach problem (asked by Ringel in 1967) by Glock, Joos, Kim, Kuhn and Osthus. Here we present a new and significantly shorter proof of the blow-up lemma for approximate decompositions. In fact, we prove a more general theorem that yields packings with stronger quasirandom properties so that it can be combined with Keevashs results on designs to obtain results of the following form. For all $varepsilon>0$, $rin mathbb{N}$ and all large $n$ (such that $r$ divides $n-1$), there is a decomposition of $K_n$ into any collection of $r$-regular graphs $H_1,ldots,H_{(n-1)/r}$ on $n$ vertices provided that $H_1,ldots,H_{varepsilon n}$ contain each at least $varepsilon n$ vertices in components of size at most $varepsilon^{-1}$.
We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to longstanding decomposition problems. For instance, our results imply the following. Let $G$ be a quasi-random $n$-vertex graph and
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlos, Sarkozy and Szemeredi that applies to almost optimally bounded colourings. A corollary of this
The blow-up lemma states that a system of super-regular pairs contains all bounded degree spanning graphs as subgraphs that embed into a corresponding system of complete pairs. This lemma has far-reaching applications in extremal combinatorics. We
A short proof is given that the graphs with proper interval representations are the same as the graphs with unit interval representations.
A graph is strongly perfect if every induced subgraph H has a stable set that meets every maximal clique of H. A graph is claw-free if no vertex has three pairwise non-adjacent neighbors. The characterization of claw-free graphs that are strongly per