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arXiv · 1410.6806

Gröbner Bases and Nullstellensätze for Graph-Coloring Ideals

Jesús A. De Loera·Susan Margulies·Michael Pernpeintner·Eric Riedl·David Rolnick·Gwen Spencer·Despina Stasi·Jon Swenson
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Abstract

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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Jesús A. De Loera, Susan Margulies, Michael Pernpeintner, Eric Riedl, David Rolnick, Gwen Spencer, Despina Stasi, Jon Swenson. 2014-10-24. Gröbner Bases and Nullstellensätze for Graph-Coloring Ideals. https://arxiv.org/abs/1410.6806

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