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arXiv · 2312.06781

Hamilton cycles in random digraphs with minimum degree at least one

Colin Cooper·Alan Frieze
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Abstract

We study the existence of a directed Hamilton cycle in random digraphs with $m$ edges where we condition on minimum in- and out-degree at least one. Denote such a random graph by $D_{n,m}^{(δ\geq1)}$. We prove that if $m=\tfrac n2(\log n+2\log\log n+c_n)$ then \[ \lim_{n\to\infty}\Pr(D_{n,m}^{(δ\geq1)}\text{ is Hamiltonian})=\begin{cases}0&c_n\to-\infty.\\e^{-e^{-c}/4}&c_n\to c.\\1&c_n\to\infty.\end{cases} \]

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Colin Cooper, Alan Frieze. 2025-06-16. Hamilton cycles in random digraphs with minimum degree at least one. https://arxiv.org/abs/2312.06781

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