Triangle Factors In Sparse Pseudo-Random Graphs

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).

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Correspondence to Michael Krivelevich*.

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* Supported by a USA-Israeli BSF grant, by a grant from the Israel Science Foundation and by a Bergmann Memorial Award.

† Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey.

‡ Research supported in part by NSF grant DMS 99-70270 and by the joint Berlin/Zurich graduate program Combinatorics, Geometry, Computation, financed by the German Science Foundation (DFG) and ETH Zürich.

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Krivelevich*, M., Sudakov†, B. & Szabó‡, T. Triangle Factors In Sparse Pseudo-Random Graphs. Combinatorica 24, 403–426 (2004).

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Mathematics Subject Classification (2000):

  • 05C70
  • 05C80