Chromatic polynomials and order ideals of monomials

One expansion of the chromatic polynomial n(G,x) of a graph G relies on spanning trees of a graph. In fact, for a connected graph G of order n, one can express n(G,x) = (1 )“-‘x cyi’=;’ ti (1 -x)‘, where ti is the nun&r of spanning trees with external activity 0 and internal activity i. Moreover, it is known (via commutative ring theory) that ti arises as the number of monomials of degree n i 1 in a set of monomials closed under division. We describe here how to explicitly carry out this construction algebraically. We also apply this viewpoint to prove a new bound for the roots of chromatic polynomials. @ 1998 Elsevier Science B.V. All rights reserved Keywork: Chromatic polynomial; Graph; Tree basis; Broken circuit complex; Order ideal of monomials; Griibner basis