Binomial edge ideals of Clutters

Abstract

In this paper, we introduce the notion of binomial edge ideals of a clutter and obtain results similar to those obtained for graphs by Rauf & Rinaldo in [8]. We also answer a question posed in their paper.

Construction of Cohen-Macaulay binomial edge ideals of graphs and clutters

“…Similar to binomial edge ideals of graphs, we have given a notion of binomial edge ideals for clutters in [7]. In this section we recall some definitions and notations from [7].…”

“…Similar to binomial edge ideals of graphs, we have given a notion of binomial edge ideals for clutters in [7]. In this section we recall some definitions and notations from [7]. We then define the notion of a closed clutter in our context and generalize the construction of Cohen-Macaulay binomial edge ideals for clutters as we have done for graphs in the previous section.…”

“…Our primary aim in this paper is to give a construction to create a graph [G] from an arbitrary graph G, such that the binomial edge ideal J [G] corresponding to the graph [G] is Cohen-Macaulay and J [G] contains the binomial edge ideal J G of G. In this construction we do not change the vertex set (i.e., the base polynomial ring) and carry out the construction by adding suitable edges only. We also show that this construction can be carried out in the more general context of binomial edge ideals of clutters, which has been introduced in [7]. The question that emerges from our construction is that what is the optimal number of steps required to construct the graph [G] from the graph G?…”

“…[7]). For a clutter C, let G C be the graph with the vertex setV (G C ) = V (C) and the edge set E(G C ) = {{i, j} | ∃ e ∈ E(C); {i, j} ⊆ e}.…”