A Weighted Graph Problem from Commutative Algebra

We give an especially simple proof of a theorem in graph theory that forms the key part of the solution to a problem in commutative algebra, on how to characterize the integral closure of a polynomial ring generated by quadratic monomials. Forms of this theorem were proved independently by two groups of researchers, who submitted their results [3, 5] two months apart, in 1997. The techniques of [3] are elementary, but involve a detailed study of aspects of integral polytopes in spaces R, while those of [5] are more eclectic, relying in part on technical aspects of commutative algebra. Our focus is on the problem as one that merely concerns weighted graphs, but motivating background from commutative algebra will first be sketched. Let R = k[x1, . . . , xn] be a commutative polynomial ring over a field k, and consider subrings F and ideals I of R generated by certain sets of polynomials. Related objects such as the integral closure F̄ of F (which can be regarded as a subring of R), and integral closures of powers and symbolic powers of I, are of interest in algebraic geometry. As can be seen from the monograph [6], a rich and still developing theory, involving for example hypergraphs and polyhedral combinatorics, arises from the study of those F and I generated by monomials. In this case, it is easy to prove that F̄ is the k-vector subspace of R generated by the monomials f ∈ √ F (some power f lies in F ) that are quotients f = g/h of monomials in F . Specializing further, to quadratic monomials xixj or x 2 i , permits more detailed results, such as those of [4], stated in terms of the associated graphs G on {x1, . . . , xn} whose edges correspond to the given generators of F . The characterization of F̄ above, applied to the present special case, translates at once to the purely graph-theoretic problem below. Although both [3] and [5] already provide solutions to the problem, we believe that the following short approach, along with the algorithm described implicitly, is worth recording. Throughout G = (V, E) is a graph, where loops but not multiple edges are allowed. Standard graph-theoretic terminology will be presupposed. Also define AV (G) to be the abelian group freely generated by V . It is convenient to use the natural embedding of AV (G) in the vector space of rational-valued weight functions f , of finite support, on the vertices of G. Subsets X of V will be identified with their characteristic functions fX in AV (G). For an ordinary edge e = {v, w}, fe is v +w, but let fe be 2v when e is a loop at the vertex v. Let AE(G) denote the subgroup of AV (G) generated by the edges of G. The problem referred to above is to describe those elements f of AE(G) for which some mf (m ∈ Z) is a positive integral linear combination of edges. Such f clearly form an additive subsemigroup P (G) of AE(G) that contains all edges of G. Following [4, Def. 6.7], an H-configuration in G is an induced subgraph H consisting of two disjoint odd cycles, perhaps loops, that lie in the same connected component of G, but which cannot be connected by adding just one edge of G. (The same configurations appear in [3]; they are more restrictive than the bowties of [5].) Here 2fH , but not fH , is a positive integral linear combination of edges. One can also see that fH is a weighted sum, with coefficients ±1, of edges, so fH ∈ P (G). With these definitions, the graph-theoretical theorem can be simply stated.