The evolution of the cube

Abstract

This chapter focuses on the study of random subgraphs of Cn. The probability that Cnp; contains a component with at least 2 and at most n/2 vertices tends to 0. Cnp is considered be a random subgraph obtained by choosing the edges of Cn independently of each other and with probability p, the probability that Cn,p is connected is discussed. The problem is particularly interesting because of a curious “double jump” at p = 1/2, and it is proved that the probability of C; being connected tends to e-1 for p=4. This double jump is reminiscent of the double jump in the size of the largest component of a random graph. A slight extension of this result is proved: by concentrating on a small neighbourhood of p=1/2 and shows the existence of a continuous probability distribution. © 1983, North-Holland Publishing Company

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North-Holland Mathematics Studies

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