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We revisit a well-known family of polynomial ideals encoding the problem ofgraph-$k$-colorability. Our paper describes how the inherent combinatorialstructure of the ideals implies several interesting algebraic properties.Specifically, we provide lower bounds on the difficulty of computing Gr\"obnerbases and Nullstellensatz certificates for the coloring ideals of generalgraphs. For chordal graphs, however, we explicitly describe a Gr\"obner basisfor the coloring ideal, and provide a polynomial-time algorithm.

Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals