Cohen-Macaulay rings associated with digraphs (Q1270082)

Let \(G\) be a digraph, that is, directed graph. A Hamilton cycle of \(G\) is a cycle containing all the vertices of \(G\). With \(G\) is associated a Cohen-Macaulay ring \(A/I(V(G))\) where \(A= k[X_1,\dots, X_r]\), \(k\) a field, \(r\) is the number of vertices of \(G\), and \(I(V(G))\) is a certain ideal of \(A\) generated by polynomials corresponding to a certain submodule \(V(G)\) of \(\mathbb{Z}^r\). The main theorem proved is that for a strongly connected digraph \(G\), the number of Hamilton cycles of \(G\) is less than or equal to the Macaulay type associated with the monoid ring \(A/I(V(G))\). Sufficient conditions are also given for them to be equal.