CHAIN LENGTHS IN CERTAIN RANDOM DIRECTED-GRAPHS

We study the random directed graph with vertex set {1,...,n} in which the directed edges (i, j) occur independently with probability c(n)/n for i < j and probability zero for i greater-than-or-equal-to j. Let M(n) (resp., L(n)) denote the length of the longest path (resp., longest path starting from vertex 1). When c(n) is bounded away from 0 and infinity as n --> infinity, the asymptotic behavior of M(n) was analyzed in previous work of the author and J. E. Cohen. Here, all restrictions on c(n) are eliminated and the asymptotic behavior of L(n) is also obtained. In particular, if c(n)/ln(n)--> infinity while c(n)/n-->0, then both M(n)/c(n) and L(n)/c(n) are shown to converge in probability to the constant e.