A New Proof of the $H$-Coloring Dichotomy

  title={A New Proof of the \$H\$-Coloring Dichotomy},
  author={M. Siggers},
  journal={SIAM Journal on Discrete Mathematics},
  • M. Siggers
  • Published 2009
  • Mathematics
  • SIAM Journal on Discrete Mathematics
In this paper, we present a new proof of the $H$-coloring dichotomy, which was first proved by Hell and Nesetril in 1990, and then was reproved by Bulatov in 2005. Our proof is much shorter than the original proof and avoids the algebraic machinery of Bulatov's proof. 
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  • 2005
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  • 1998
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  • Combin. Theory B 48
  • 1990
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  • 2009