Cohen-Macaulay rings in network reliability

For any simplicial complex Delta and field K, one can associate a graded K-algebra K[Delta] (the Stanley-Reisner ring). For certain Delta and K, the Stanley-Reisner rings have a homogeneous system of parameters, Theta, such that K[Delta]/[Theta] is finite-dimensional, and coefficients of its Hilbert series are the h-vector of Delta. The previous constructions of Theta were noncombinatorial. In the special. case of cographic matroids, we give (for any field K) a combinatorial description of a homogeneous system of parameters in terms of the graph structure, as well as an explicit basis for the resulting quotient algebra. The results have applications to a central problem of reliability, namely the association of a multicomplex to a connected graph, such that the reliability is a simple function of the rank numbers.