We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire’s splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bi...

Let [Formula: see text] be a connected graph on the vertex set text]. Then In this paper, we prove that if is unicyclic graph, then depth of bounded below by Also, characterize with and We compute one distinguished extremal Betti numbers If obtained attaching whiskers at some vertices cycle length text], show Furthermore, each these cases, classify uniqueness number graphs.

Let G be a graph on [ n ] and J the binomial edge ideal of in polynomial ring S = K x 1 , … y . In this paper we investigate some topological properties poset associated to minimal primary decomposition We show that admits specific subposets which are contractible. This turn, provides interesting algebraic consequences. particular, characterize all graphs for depth / 4