Abstract:Loebl, Komlós and Sós 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 Szemerédi'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 (… Show more
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