The goal of the paper is to initiate research towards a general, Blow-up Lemma type embedding statement for pseudo-random graphs with sublinear degrees. In particular, we show that if the second eigenvalue λ of a d-regular graph G on 3n vertices is at most cd 3/n 2 log n, for some sufficiently small constant c > 0, then G contains a triangle factor. We also show that a fractional triangle factor already exists if λ < 0.1d 2/n. The latter result is seen to be best possible up to a constant factor, for various values of the degree d = d(n).
Mathematics Subject Classification (2000):05C70 05C80
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