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In this paper we consider the problem of embedding almost‐spanning, bounded degree graphs in a random graph. In particular, let    Δ  ⩾  5   ,    ε  >  0   , and let   H   be a graph on    (  1  −  ε  )  n    vertices and with maximum degree   Δ  . We show that a random graph    G   n  ,  pwith high probability contains a copy of   H  , provided that    p  ≫    (   n   −  1log   1  /  Δ    n  )    2  /  (  Δ  +  1  ). Our assumption on   p   is optimal up to the    p  o  l  y  l  o  g    factor. We note that this    p  o  l  y  l  o  g    term matches the conjectured threshold for the spanning case.

Embedding large graphs into a random graph