Packing d-degenerate graphs

We study packings of graphs with given maximal degree. We shall prove that the (hitherto unproved) Bollobás–Eldridge–Catlin Conjecture holds in a considerably stronger form if one of the graphs is d-degenerate for d not too large: if d , Δ 1 , Δ 2 ⩾ 1 and n > max { 40 Δ 1 ln Δ 2 , 40 d Δ 2 } then a d-degenerate graph of maximal degree Δ 1 and a graph of order n and maximal degree Δ 2 pack. We use this result to show that, for d fixed and n large enough, one can pack n 1500 d 2 arbitrary d-degenerate n-vertex graphs of maximal degree at most n 1000 d ln n .