Recognizing Graph Theoretic Properties with Polynomial Ideals

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry. The first and third author are partially supported by NSF grant DMS-0914107 and an IBM OCR award. The second author is partially supported by an NSA Young Investigator Grant and an NSF AllInstitutes Postdoctoral Fellowship administered by the Mathematical Sciences Research Institute through its core grant DMS-0441170. The fourth author is partially supported by NSERC Postgraduate Scholarship 281174. the electronic journal of combinatorics 17 (2010), #R114 1