Summary: Spanning subgraphs of Random Graphs
(A research problem)
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
Department of Mathematics, University of Illinois, Urbana, IL 61801 USA, and
Mathematical Institute of the Hungarian Academy, POB 127, Budapest 1364, Hungary
We propose a problem concerning the determination of the threshold function for the edge
probability that guarantees, almost surely, the existence of various sparse spanning subgraphs
in random graphs. We prove some bounds and demonstrate them in the cases of a d-cube and
a two dimensional lattice.
B. Bollob´as (cf. e.g., ) raised the following problem:
Let G be a random graph with n = 2d vertices, in which each edge is taken randomly and indepen-
dently with probability p = 1 - , where is a positive small constant. Is it true that for d > d()
almost surely G contains a copy of the d-cube, Qd? Note that Qd has 2d-1d = O(n log n) edges,
and is thus a relatively sparse graph.