On Two Conjectures on Packing of Graphs

In 1978, Bollobás and Eldridge [5] made the following two conjectures. (C1) There exists an absolute constant c > 0 such that, if k is a positive integer and G1 and G2 are graphs of order n such that ∆(G1),∆(G2) n− k and e(G1), e(G2) ckn, then the graphs G1 and G2 pack. (C2) For all 0 < α < 1/2 and 0 < c < √ 1/8, there exists an n0 = n0(α, c) such that, if G1 and G2 are graphs of order n > n0 satisfying e(G1) αn and e(G2) c √ n3/α, then the graphs G1 and G2 pack. Conjecture (C2) was proved by Brandt [6]. In the present paper we disprove (C1) and prove an analogue of (C2) for 1/2 α < 1. We also give sufficient conditions for simultaneous packings of about √ n/4 sparse graphs.