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Hamiltonicity in random directed graphs is born resilient

Part of: Graph theory

Published online by Cambridge University Press:  24 June 2020

Richard Montgomery
Richard Montgomery*
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
*
Email: [email protected]
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Abstract

Let $\{D_M\}_{M\geq 0}$ be the n-vertex random directed graph process, where $D_0$ is the empty directed graph on n vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $$\varepsilon > 0$$ , we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $1/2-\varepsilon$ of both the in- and out-edges incident to each vertex are removed. We say such a directed graph is $(1/2-\varepsilon)$ -resiliently Hamiltonian. Furthermore, for each $\varepsilon > 0$ , we show that, almost surely, each directed graph $D_M$ in the sequence is not $(1/2+\varepsilon)$ -resiliently Hamiltonian.

This improves a result of Ferber, Nenadov, Noever, Peter and Škorić who showed, for each $\varepsilon > 0$ , that the binomial random directed graph $D(n,p)$ is almost surely $(1/2-\varepsilon)$ -resiliently Hamiltonian if $p=\omega(\log^8n/n)$ .

MSC classification

Primary: 05C20: Directed graphs (digraphs), tournaments
Secondary: 05C80: Random graphs 05C38: Paths and cycles 05C45: Eulerian and Hamiltonian graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 6 , November 2020 , pp. 900 - 942
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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