Binomial Edge Ideals and Related Ideals

In this chapter we consider classes of binomial ideals which are naturally attached to finite simple graphs. The first of these classes are the binomial edge ideals. These ideals may also be viewed as ideals generated by a subset of 2-minors of a (2 × n)-matrix of indeterminates. Their Grobner bases will be computed. Graphs whose binomial edge ideals have a quadratic Grobner basis are called closed graphs. A full classification of closed graphs is given. For an arbitrary graph the initial ideal of the binomial edge ideal (for a suitable monomial order) is a squarefree monomial ideal. This has the pleasant consequence that the binomial edge ideal itself is a radical ideal. Its minimal prime ideals are determined in terms of cut point properties of the underlying graph. Based on this information, the closed graphs whose binomial edge ideal is Cohen–Macaulay are classified. In the subsequent sections, the resolution of binomial edge ideals is considered and a bound for the Castelnuovo–Mumford regularity of these ideals is given. Finally, the Koszul property of binomial edge ideals is studied. Intimately related to binomial edge ideals are permanental edge ideals and Lovasz, Saks, and Schrijver edge ideals. Their primary decomposition will be studied.