Article
The ErdősSós conjecture for graphs of girth 5
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
Discrete Mathematics
(Impact Factor: 0.57).
04/1996;
150(13):411414.
DOI: 10.1016/0012365X(95)00207D

Article: On the Loebl–Komlós–Sós conjecture
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ABSTRACT: The Loebl–Komlós–Sós conjecture says that any graph G on n vertices with at least half of vertices of degree at least k contains each tree of size k. We prove that the conjecture is true for paths as well as for large values of k(k ≥ n  3). © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 269–276, 2000Journal of Graph Theory 08/2000; 34(4):269276. · 0.67 Impact Factor 
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ABSTRACT: In a series of four papers we prove the following relaxation of the LoeblKomlosSos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemer\'edi regularity lemma: we decompose the graph $G$, find a suitable combinatorial structure inside the decomposition, and then embed the tree $T$ into $G$ using this structure. Since for sparse graphs $G$, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three followup papers, we find a combinatorial structure suitable inside the decomposition, which we then use for embedding the tree. 
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ABSTRACT: Loebl, Koml\'os and S\'os conjectured that every $n$vertex graph $G$ with at least $n/2$ vertices of degree at least $k$ contains each tree $T$ of order $k+1$ as a subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of $k$. For our proof, we use a structural decomposition which can be seen as an analogue of Szemer\'edi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of $G$ to embed a given tree $T$. The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050].
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