On Two Conjectures on Packing of Graphs

BÉLA BOLLOBÁS, ALEXANDR KOSTOCHKA, KITTIKORN NAKPRASIT
2005 Combinatorics, probability & computing  
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 G 1 and G 2 are graphs of order n such that ∆(G 1 ), ∆(G 2 ) n − k and e(G 1 ), e(G 2 ) ckn, then the graphs G 1 and G 2 pack. (C2) For all 0 < α < 1/2 and 0 < c < 1/8, there exists an n 0 = n 0 (α, c) such that, if G 1 and G 2 are graphs of order n > n 0 satisfying e(G 1 ) αn and e(G 2 ) c n 3 /α, then the graphs G 1 and G 2 pack.
more » ... d G 2 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.
doi:10.1017/s0963548305006887 fatcat:zshbdqnvubdadiz7csfdjm5pvq