Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals
  We revisit a well-known family of polynomial ideals encoding the problem of graph-$k$-colorability. Our paper describes how the inherent combinatorial structure of the ideals implies several interesting algebraic properties. Specifically, we provide lower bounds on the difficulty of computing Gr\"obner bases and Nullstellensatz certificates for the coloring ideals of general graphs. For chordal graphs, however, we explicitly describe a Gr\"obner basis for the coloring ideal, and provide a polynomial-time algorithm. 
Jes\'us A. De Loera, Susan Margulies, Michael Pernpeintner, Eric   Riedl, David Rolnick, Gwen Spencer, Despina Stasi, Jon Sw...
cs.SC cs.CC math.AC math.AG math.CO