Regular edges, matchings and Hilbert series

Abstract When I is the edge ideal of a graph G , we use combinatorial properties, particularly property P on connectivity of neighbors of an edge, to classify when a binomial sum of vertices is a regular element on R / I ( G ). Under a mild separability assumption, we identify when such elements can be combined to form a regular sequence. Using these regular sequences, we show that the Hilbert series and corresponding h -vector can be calculated from a related graph using a simplified calculation on the f -vector, or independence vector, of the related graph. In the case when the graph is Cohen–Macaulay with a perfect matching of regular edges satisfying the separability criterion, the h -vector of R / I ( G ) will be precisely the f -vector of the Stanley–Reisner complex of a graph with half as many vertices as G .