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,∆1,∆2 ≥ 1 and n > max{40∆1 ln ∆2, 40d∆2} then a d-degenerate graph of maximal degree ∆1 and a graph of order n and maximal degree ∆2 pack. We use this result to show that, for d fixed and n large enough, one can pack n 1500d2 arbitrary d-degenerate n-vertex graphs of maximal degree at most n 1000d lnn .