  This article is built upon three main ideas. First, for a class of monomial
ideals, it is proven that the multiplicity of an ideal equals the number of
realizations of its codimension (an intuitive concept that we define later).
Next, for an arbitrary graph G, we construct a monomial ideal M_G, and show
that the chromatic number of G is equal to the codimension of M_G. Finally, for
a class of graphs, we give a formula that computes the chromatic polynomial of
G, evaluated at the chromatic number of G, in terms of the codimension and
multiplicity of M_G. In particular, the formula applies to all graphs
satisfying the Erdos-Faber-Lov\'asz conjecture.
