Packing d-degenerate graphs

AbstractWe 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,Δ1,Δ2⩾1 and n>max{40Δ1lnΔ2,40dΔ2} then a d-degenerate graph of maximal degree Δ1 and a graph of order n and maximal degree Δ2 pack. We use this result to show that, for d fixed and n large enough, one can pack n1500d2 arbitrary d-degenerate n-vertex graphs of maximal degree at most n1000dlnn