Gröbner Bases and Nullstellensätze 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 Gröbner bases and Nullstellensatz certificates for the coloring ideals of general graphs. For chordal graphs, however, we explicitly describe a Gröbner basis for the coloring ideal, and provide a polynomial-time algorithm.
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