The Coloring Ideal and Coloring Complex of a Graph

Let G be a simple graph on d vertices. We define a monomial ideal K in the Stanley-Reisner ring A of the order complex of the Boolean algebra on d atoms. The monomials in K are in one-to-one correspondence with the proper colorings of G. In particular, the Hilbert polynomial of K equals the chromatic polynomial of G. The ideal K is generated by square-free monomials, so A/K is the Stanley-Reisner ring of a simplicial complex C . The h-vector of C is a certain transformation of the tail T (n) = nd − χ(n) of the chromatic polynomial χ of G. The combinatorial structure of the complex C is described explicitly and it is shown that the Euler characteristic of Cequals the number of acyclic orientations of G.