Embedding graphs with bounded degree in sparse pseudorandom graphs

In this paper, we show the equivalence of somequasi-random properties for sparse graphs, that is, graphsG with edge densityp=|E(G)|/(2n)=o(1), whereo(1)→0 asn=|V(G)|→∞. Our main result (Theorem 16) is the following embedding result. For a graphJ, writeNJ(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 setdH=max{δ(J):J⊆H}. Moreover, putDH=min{2dH,Δ(H)}. LetC>1 be a fixed constant and supposep=p(n)≫n−1DH. We show that ifG is such thatdegG(x)≤Cpn for allx∈V(G),for all 2≤r≤DH and for all distinct verticesx1, ...,xr ∈V(G),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left| {N_G (x_1 ) \cap \cdots \cap N_G (x_r )} \right| \leqslant Cnp^r $$ \end{document},for all but at mosto(n2) pairs {x1,x2} ⊆V(G),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\| {N_G (x_1 ) \cap N_G (x_2 )\left| { - np^2 } \right| = o(np_2 )} \right.$$ \end{document}, then the number of labeled copies ofH inG is\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$N(H,G_n ) = (1 + o(1))n^{\left| {V(H)} \right|} p^{\left| {E(H)} \right|} $$ \end{document}.