# 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} }

- Published 1998 in Combinatorics, Probability & Computing

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.Â

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