Viewing counting polynomials as Hilbert functions via Ehrhart theory

Abstract

Steingrímsson (2001) showed that the chromatic polynomial of a graph is the Hilbert function of a relative Stanley- Reisner ideal. We approach this result from the point of view of Ehrhart theory and give a sufficient criterion for when the Ehrhart polynomial of a given relative polytopal complex is a Hilbert function in Steingrímsson's sense. We use this result to establish that the modular and integral flow and tension polynomials of a graph are Hilbert functions. © 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.