On Two Conjectures on Packing of Graphs

  • Authors:

  • Béla Bollobás;Alexandr Kostochka;Kittikorn Nakprasit

  • Affiliations:

  • University of Memphis, Memphis, TN 38152, USA and Trinity College, Cambridge CB2 1TQ, UK (e-mail: bollobas@msci.memphis.edu);University of Illinois, Urbana, IL 61801, USA and Institute of Mathematics, Novosibirsk 630090, Russia (e-mail: kostochk@math.uiuc.edu);University of Illinois, Urbana, IL 61801, USA (e-mail: nakprasi@math.uiuc.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2005

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Abstract

In 1978, Bollobás and Eldridge [5] made the following two conjectures. (C1) There exists an absolute constant $c0$ such that, if k is a positive integer and $G_1$ and $G_2$ are graphs of order n such that $\Delta(G_1),\Delta(G_2)\leq n-k$ and $e(G_1),e(G_2)\leq ck n$, then the graphs $G_1$ and $G_2$ pack. (C2) For all $0n_0$ satisfying $e(G_1)\leq \alpha n$ and $e(G_2)\leq c\sqrt{n^3/ \alpha}$, then the graphs $G_1$ and $G_2$ pack. Conjecture (C2) was proved by Brandt [6]. In the present paper we disprove (C1) and prove an analogue of (C2) for $1/2\leq \alpha