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)|/( ⁿ₂ )=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

• (i) deg _G (x)≤C _pn for allx∈V(G),

• (ii) for all 2≤r≤D _H and for all distinct verticesx ₁, ...,x _r ∈V(G),

  $$\left| {N_G (x_1 ) \cap \cdots \cap N_G (x_r )} \right| \leqslant Cnp^r $$

  ,

• (iii) for all but at mosto(n ²) pairs {x ₁,x ₂} ⊆V(G),

  $$\left\| {N_G (x_1 ) \cap N_G (x_2 )\left| { - np^2 } \right| = o(np_2 )} \right.$$

  , then the number of labeled copies ofH inG is

  $$N(H,G_n ) = (1 + o(1))n^{\left| {V(H)} \right|} p^{\left| {E(H)} \right|} $$

  .

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.