Colouring Random 4-Regular Graphs

Authors:
Lingsheng Shi;Nicholas Wormald
Affiliations:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, Canada N2L 3G1 (e-mail: lshi@math.tsinghua.edu.cn, nwormald@uwaterloo.ca);Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, Canada N2L 3G1 (e-mail: lshi@math.tsinghua.edu.cn, nwormald@uwaterloo.ca)
Venue:
Combinatorics, Probability and Computing
Year:
2007

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Abstract

We show that a random 4-regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. The proof uses an efficient algorithm which a.a.s. 3-colours a random 4-regular graph. The analysis includes use of the differential equation method, and exponential bounds on the tail of random variables associated with branching processes. A substantial part of the analysis applies to random $d$-regular graphs in general.