Embedding graphs with bounded degree in sparse pseudorandom graphs

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)\ --> infinity. Our main result (Theorem 16) is the following embedding result. For a graph J, write N (J)(x) for the neighborhood of the vertex x in J, and let delta (J) and Delta (J) be the minimum and the maximum degree in J. Let H be a triangle-free graph and set d(H) = max {delta(J): J subset of or equal to H}. Moreover, put D-H = min {2d(H), Delta(H)}. Let C > 1 be a fixed constant and suppose p = p(n) much greater than n(-1/DH). We show that if G is such that (i) deg(G)(x) less than or equal to Cpn for all x is an element of V(G), (ii) for all 2 less than or equal to r less than or equal to D-H and for all distinct vertices x(1), . . . , x(r) is an element of V(G), \N-G (x(1)) boolean AND (. . .) boolean AND N-G (x (r))\ less than or equal to Cnp(r), (iii) for all but at most o(n(2)) pairs {x(1), x(2)} subset of or equal to V(G), parallel to(N) over bar (G) (x(1)) boolean AND N-G (x(2))\ - np(2)\ = 0(np(2)), then the number of labeled copies of H in G is N(H, G(n)) = (1 + o(1))n(\V(H)\ p \E(H)\). Moreover, 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.