# Discrete Mathematics & Theoretical Computer Science

## Volume 5 n° 1 (2002), pp. 205-226

author: | Nikolaos Fountoulakis and Colin McDiarmid |
---|---|

title: | Upper bounds on the non-3-colourability threshold of random graphs |

keywords: | sparse random graphs, 3-colourability, thresholds |

abstract: |
We present a full analysis of the expected number of
`rigid' 3-colourings of a sparse random graph. This shows
that, if the average degree is at least 4.99, then as n
-> ∞ the expected number of such
colourings tends to 0 and so the probability that the
graph is 3-colourable tends to 0. (This result is tight,
in that with average degree 4.989 the expected number
tends to ∞.) This bound appears independently in
Kaporis et al: A Note on the Non-Colourability Threshold of a Random Graph. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989. If your browser does not display the abstract correctly (because of the different mathematical symbols) you can look it up in the PostScript or PDF files. |

reference: | Nikolaos Fountoulakis and Colin McDiarmid (2002),
Upper bounds on the non-3-colourability threshold of random graphs,
Discrete Mathematics and Theoretical Computer Science 5, pp. 205-226 |

bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |

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Automatically produced on Mon Nov 4 10:03:11 CET 2002 by ifalk