### 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) |
---|---|

Pages (from-to) | 243-253 |

Number of pages | 11 |

Journal | Random Structures and Algorithms |

Volume | 3 |

Issue number | 3 |

DOIs | |

State | Published - 1992 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

**Chain Lengths in Certain Random Directed Graphs.** / Newman, Charles.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 3, no. 3, pp. 243-253. https://doi.org/10.1002/rsa.3240030304

}

TY - JOUR

T1 - Chain Lengths in Certain Random Directed Graphs

AU - Newman, Charles

PY - 1992

Y1 - 1992

N2 - We study the random directed graph with vertex set {1, …, n} in which the directed edges (i, j) occur independently with probability cn/n for in (resp., Ln) denote the length of the longest path (resp., longest path starting from vertex 1). When cn is bounded away from 0 and ∞ as n→∞, the asymptotic behavior of Mn was analyzed in previous work of the author and J. E. Cohen. Here, all restrictions on cn are eliminated and the asymptotic behavior of Ln is also obtained. In particular, if cn/ln(n)→∞ while cn/n→0, then both Mn/cn and Ln/cn are shown to converge in probability to the constant e.

AB - We study the random directed graph with vertex set {1, …, n} in which the directed edges (i, j) occur independently with probability cn/n for in (resp., Ln) denote the length of the longest path (resp., longest path starting from vertex 1). When cn is bounded away from 0 and ∞ as n→∞, the asymptotic behavior of Mn was analyzed in previous work of the author and J. E. Cohen. Here, all restrictions on cn are eliminated and the asymptotic behavior of Ln is also obtained. In particular, if cn/ln(n)→∞ while cn/n→0, then both Mn/cn and Ln/cn are shown to converge in probability to the constant e.

UR - http://www.scopus.com/inward/record.url?scp=84990739017&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84990739017&partnerID=8YFLogxK

U2 - 10.1002/rsa.3240030304

DO - 10.1002/rsa.3240030304

M3 - Article

VL - 3

SP - 243

EP - 253

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -