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.
© 2010 Jesús A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar
De Loera, J., Hillar, C., Malkin, P., and Omar, M. Recognizing Graph Theoretic Properties with Polynomial Ideals., Electronic Journal of Combinatorics. Vol 17, R114 (2010).