Embedding graphs with bounded degree in sparse pseudorandom graphs

Abstract

In this paper, we show the equivalence of somequasi-random properties for sparse graphs, that is, graphsG with edge densityp=|E(G)|/( 2 )=o(1), whereo(1)→0 asn=|V(G)|→∞. Our main result (Theorem 16) is the following embedding result. For a graphJ, writeN J(x) for the neighborhood of the vertexx inJ, and letδ(J) andΔ(J) be the minimum and the maximum degree inJ. LetH be atriangle-free graph and setd H=max{δ(J):J⊆H}. Moreover, putD H=min{2d H,Δ(H)}. LetC>1 be a fixed constant and supposep=p(n)≫n −1 D H. We show that ifG is such that Moreover, we discuss a setting under which an arbitrary graphH (not necessarily triangle-free) can be embedded inG. We also present an embedding result for directed graphs.