Splittings of Binomial Edge Ideals

Abstract

Consider a finite simple graph G. One can associate an ideal to the edges of this graph, called its binomial edge ideal. Many homological invariants, such as the Betti numbers, Castelnuovo-Mumford regularity (reg) and the projective dimension (pd) of these ideals are widely studied. For binomial edge ideals of graphs, these invariants are often intimately related to graph-theoretic notions such as connectivity, free vertices and so on. In this thesis, we study the method of Betti splittings applied to binomial edge ideals. We give some examples of Betti splittings and introduce the notion of a partial Betti splitting. We demonstrate that breaking off a vertex and its incident edges from the graph results in a partial Betti splitting of the associated binomial edge ideal. A similar study is also done to obtain a partial Betti splitting for the initial ideal of a binomial edge ideal. We also prove new bounds for some homological invariants of the binomial edge ideal and explore some of their implications.