## Models of Random Regular Graphs (1999)

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Venue: | In Surveys in combinatorics |

Citations: | 157 - 33 self |

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@INPROCEEDINGS{Shi99modelsof,

author = {Lingsheng Shi and Nicholas Wormald},

title = {Models of Random Regular Graphs},

booktitle = {In Surveys in combinatorics},

year = {1999},

pages = {239--298},

publisher = {University Press}

}

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### Abstract

In a previous paper we showed that a random 4-regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6-regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random d-regular graph for other d between 5 and 10 inclusive is a.a.s. restricted to a range of two integer values: {3, 4} for d = 5, {4, 5} for d = 7, 8, 9, and {5, 6} for d = 10. The proof uses efficient algorithms which a.a.s. colour these random graphs using the number of colours specified by the upper bound. These algorithms are analysed using the differential equation method, including an analysis of certain systems of differential equations with discontinuous right hand sides. 1

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Citation Context ...y further, by working with systems of equations which specify right hand derivatives only. The derivatives in the natural setting are sometimes discontinuous at phase changes. In the past ([9], [18], =-=[19]-=- and [21] for example), the associated solution functions were cut apart at the phase changes so that within each phase the derivative is smooth. Due to the difficulty of identifying the phase changes... |

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Citation Context ...per bounds, we obtain algorithms which a.a.s. colour these random graphs with the number of colours given by the upper bound. This compares with a recent non-algorithmic result of Achlioptas and Naor =-=[2]-=- that for the common random graph model G(n, d/n) (in which edges between the n vertices are chosen independently at random with probability d/n), the chromatic number is, when d is fixed, concentrate... |

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Citation Context ... ̷Luczak [11] showed that χ is a.a.s. asymptotic to d 2 log d moderately slowly, there are functions of d asymptotic to d 2 log d lower bounds on χ(G) for G ∈ Gn,d. This was extended by Cooper et al. =-=[8]-=- to d = o(n 1−η ) for all fixed η > 0 (provided d → ∞), and, earlier than that, by Krivelevich et al. [14, Theorem 2.5] to n 6/7+η ≤ d ≤ 0.9n with the formula adjusted to (1 + o(1))n/(2 log b d) where... |

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Citation Context ...ent in [16], at the end of this section. They cannot be obtained by a simple direct evaluation of the expected number of k-colourings. While the present paper was being prepared, Achlioptas and Moore =-=[1]-=- announced a proof of two things: firstly that the chromatic number of Gn,d is 2-point concentrated, so is a.a.s. k or k + 1 for some k = k(d), and secondly that if k is the smallest integer such that... |

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Citation Context ...ntegers, and on one value for d in some intervals. A number of results on the chromatic number of Gn,d have already appeared. Frieze , in the sense that as d → ∞ that are a.a.s. upper and and ̷Luczak =-=[11]-=- showed that χ is a.a.s. asymptotic to d 2 log d moderately slowly, there are functions of d asymptotic to d 2 log d lower bounds on χ(G) for G ∈ Gn,d. This was extended by Cooper et al. [8] to d = o(... |

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