Mathematics > Combinatorics
[Submitted on 17 Aug 2014 (v1), last revised 30 Nov 2015 (this version, v3)]
Title:The approximate Loebl-Komlós-Sós Conjecture I: The sparse decomposition
View PDFAbstract:In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos 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 Szemerédi 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 follow-up papers, we find a combinatorial structure suitable inside the decomposition, which we then use for embedding the tree.
Submission history
From: Jan Hladky [view email][v1] Sun, 17 Aug 2014 21:02:52 UTC (264 KB)
[v2] Tue, 14 Jul 2015 08:56:17 UTC (228 KB)
[v3] Mon, 30 Nov 2015 12:29:30 UTC (229 KB)
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