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In this paper, we show the equivalence of some quasi-random properties for sparse graphs, that is, graphs G with edge density p = |E(G)| / � � n 2 o(1), where o(1) → 0 as n = |V (G) | → ∞. In particular, we prove the following embedding result. For a graph J, write NJ(x) for the neighborhood of the vertex x in J, and let δ(J) and ∆(J) be the minimum and the maximum degree in J. Let H be a triangle-free graph and set dH = max{δ(J): J ⊂ H}. Moreover, put DH = min{2dH, ∆(H)}. Let C> 1 be a fixed constant and suppose p = p(n) ≫ n −1/DH. We show that if G is such that (i) deg G(x) ≤ Cpn for all x ∈ V (G), (ii) for all 2 ≤ r ≤ DH and for all distinct vertices x1,..., xr ∈ V (G), |NG(x1) ∩ · · · ∩ NG(xr) | ≤ Cnp r, (iii) for all but at most o(n 2) pairs {x1, x2} ⊂ V (G), � |NG(x1) ∩ NG(x2) | − np 2 � � = o(np 2), then G contains H as a subgraph. We discuss a setting under which an arbitrary graph H (not necessarily triangle-free) can be embedded in G. We also present an embedding result for directed graphs

Year: 2009

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