### Abstract

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 ⩽ 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 ∞ as n→∞, 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)→∞ 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.

Original language | English (US) |
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Pages (from-to) | 243-253 |

Number of pages | 11 |

Journal | Random Structures & Algorithms |

Volume | 3 |

Issue number | 3 |

DOIs | |

State | Published - 1992 |

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### ASJC Scopus subject areas

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics