Regular Edges, Matchings and Hilbert Series

TL;DR

This paper explores the algebraic properties of edge ideals of graphs, providing criteria for regular elements and sequences, and linking Hilbert series and h-vectors to combinatorial graph invariants, especially for Cohen-Macaulay graphs.

Contribution

It introduces combinatorial conditions for regular elements and sequences in edge ideals, and relates Hilbert series and h-vectors to simplified graph invariants, especially in Cohen-Macaulay cases.

Findings

Regular elements classified via Property P on graph connectivity.

Identification of when regular sequences can be formed from these elements.

Hilbert series and h-vectors computed from related graphs using simplified invariants.

Abstract

When $I$ is the edge ideal of a graph $G$, we use combinatorial properities, 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 $G$.