The largest component in a random subgraph of the n-cycle

Abstract

Let M denote the order of the largest component in a random subgraph H of the n-cycle C_n, where H has the same vertex set as C_n and its edge set is defined by independently selecting, with the same constant probability, each of the edges of C_n. The probability that M is equal to k is known for k=1 and for $\text{n⩾k⩾⌊}\text{n}\text{2}\text{⌋}$. Here we obtain the exact result for k=2 and comment on the cases $\text{⌊}\text{n}\text{2}\text{⌋>k>2}$.