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 n⩾k⩾⌊ n 2 ⌋ . Here we obtain the exact result for k =2 and comment on the cases ⌊ n 2 ⌋>k>2 .