Cohen–Macaulay Rings Associated with Digraphs

Abstract

Let G be a digraph, that is, a pair of sets consisting of a set of vertices Ž . and a set of directed edges for a more precise definition, see Section 3 . It is an interesting problem to know how to count the Hamilton cycles of G, that is, cycles containing all vertices of G. In this paper, we will give the upper bound of the number of them by using the theory of commutative rings. It is natural to assume that G is strongly connected, otherwise there is no Hamilton cycle. The way to give it is to associate G with a Cohen]Macaulay ring which is positively graded of dimension 1 and to compute its Macaulay type. In fact, the associated ring is a monoid ring, r Ž which is defined for a submodule of Z where Z is the ring of integers for . the definition, see Section 1 . For this reason, we will compute the Macaulay type of Cohen]Macaulay monoid ring in a special case in Section 2. w x Ž . Let A s k X , . . . , X k is a field . In Section 1, we define the 1 r Ž . r Ž . polynomial F for g Z and the ideal I V of A for a submodule V of r Ž . Ž . Z for the definition, see Section 1 . And we call the ring ArI V a monoid ring. These definitions and a few properties of this ring are found w x in 5 . The monoid ring is always a graded ring. If it is positively graded, we can use the theory of Grobner bases and show a basic lemma about Ž . Ž . generation of elements in its defining ideal I V see Proposition 1.3 . In Section 2, we only treat a special case: V is a submodule generated by r Ž . Ž . , . . . , in Z with s ) 0, s F 0 for each j / i and q ??? q 1 r i i i j 1 r s 0, where s denotes the ith entry. Further assume that the monoid ring i Ž . ArI V is positively graded of dimension 1. We will see in the next section that these conditions are satisfied if V is associated with a strongly connected digraph. Thus to investigate this case makes sense. Under these Ž . conditions ArI V is Cohen]Macaulay and we will compute its Macaulay type.