Uniquely Colourable Graphs and the Hardness of Colouring Graphs of Large Girth

@article{EmdenWeinert1998UniquelyCG,
  title={Uniquely Colourable Graphs and the Hardness of Colouring Graphs of Large Girth},
  author={Thomas Emden-Weinert and Stefan Hougardy and Bernd Kreuter},
  journal={Combinatorics, Probability & Computing},
  year={1998},
  volume={7},
  pages={375-386}
}
For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k 12(g+1) vertices whose maximal degree is at most 5k 13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G) log jGj 13 log k and maximum degree (G) 6k 13 can exist. We also study several related problems. 

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