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

Published online by Cambridge University Press:  01 December 1998

THOMAS EMDEN-WEINERT
Affiliation:
Institut für Informatik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany (e-mail: TEmdenWe@vss.comhougardy@informatik.hu-berlin.debernd.kreuter@sap-ag.de)
STEFAN HOUGARDY
Affiliation:
Institut für Informatik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany (e-mail: TEmdenWe@vss.comhougardy@informatik.hu-berlin.debernd.kreuter@sap-ag.de)
BERND KREUTER
Affiliation:
Institut für Informatik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany (e-mail: TEmdenWe@vss.comhougardy@informatik.hu-berlin.debernd.kreuter@sap-ag.de)

Abstract

For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k12(g+1) vertices whose maximal degree is at most 5k13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G)[ges ]log[mid ]G[mid ]/13logk and maximum degree Δ(G)[les ]6k13 can exist. We also study several related problems.

Type
Research Article
Copyright
1998 Cambridge University Press

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