Dirac's theorem for random graphs
Abstract
A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $\lceil n/2 \rceil$ is Hamiltonian. In this paper we extend this result to random graphs. Motivated by the study of resilience of random graph properties we prove that if $p \gg \log n /n$, then a.a.s. every subgraph of $G(n,p)$ with minimum degree at least $(1/2+o(1))np$ is Hamiltonian. Our result improves on previously known bounds, and answers an open problem of Sudakov and Vu. Both, the range of edge probability $p$ and the value of the constant 1/2 are asymptotically best possible.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2011
- DOI:
- 10.48550/arXiv.1108.2502
- arXiv:
- arXiv:1108.2502
- Bibcode:
- 2011arXiv1108.2502L
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 14 pages,1 figures