Chain Lengths in Certain Random Directed Graphs

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Abstract

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 i<j and probability zero for i ⩽ j. Let Mn (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.

Original languageEnglish (US)
Pages (from-to)243-253
Number of pages11
JournalRandom Structures & Algorithms
Volume3
Issue number3
DOIs
StatePublished - 1992

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ASJC Scopus subject areas

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

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