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Sharp threshold for embedding combs and other spanning trees in random graphs
When , the tree consists of a path containing
vertices, each of whose vertices has a disjoint path length
beginning at it. We show that, for any and , the binomial
random graph almost surely contains
as a subgraph. This improves a recent result of Kahn,
Lubetzky and Wormald. We prove a similar statement for a more general class of
trees containing both these combs and all bounded degree spanning trees which
have at least disjoint bare paths length .
We also give an efficient method for finding large expander subgraphs in a
binomial random graph. This allows us to improve a result on almost spanning
trees by Balogh, Csaba, Pei and Samotij.Comment: 20 page
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