No 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.

We generalize some results of v-number for arbitrary monomial ideals by showing that the v-number of an arbitrary monomial ideal is the same as the v-number of its polarization. We prove that the vnumber v(I(G)) of the edge ideal I(G), the induced matching number im(G) and the regularity reg, where G is either a bipartite graph, or a (C 4 , C 5 )free vertex decomposable graph, or a whisker graph. There is an open problem in [16], whether v(I) ≤ reg(R/I) + 1 for any square-free monomial ideal I. We show that v(I(G)) > reg(R/I(G)) + 1, for a disconnected graph G. We derive some inequalities of v-numbers which may be helpful to answer the above problem for the case of connected graphs. We connect v(I(G)) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that reg(R/I(G)) can be arbitrarily larger than v(I(G)). Also, we try to see how the v-number is related to the Cohen-Macaulay property of square-free monomial ideals.

For a graph G, Bolognini et al. have shown J G is strongly unmixed ⇒ J G is Cohen-Macaulay ⇒ G is accessible, where J G denotes the binomial edge ideals of G. Accessible and strongly unmixed properties are purely combinatorial. We give some motivations to focus only on blocks with whiskers for the characterization of all G with Cohen-Macaulay J G . We show that accessible and strongly unmixed properties of G depend only on the corresponding properties of its blocks with whiskers and vice versa. Also, we give an infinite class of graphs whose binomial edge ideals are Cohen-Macaulay, and from that, we classify all r-regular r-connected graphs such that attaching some special whiskers to it, the binomial edge ideals become Cohen-Macaulay. Finally, we define a new class of graphs, called strongly r-cut-connected and prove that the binomial edge ideal of any strongly r-cut-connected accessible graph having at most three cut vertices is Cohen-Macaulay.

The study of the edge ideal I(D G ) of a weighted oriented graph D G with underlying graph G started in the context of Reed-Muller type codes. We generalize a Cohen-Macaulay construction for I(D G ), which Villarreal gave for edge ideals of simple graphs. We use this construction to classify all the Cohen-Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We show that the conjecture on Cohen-Macaulayness of I(D G ), proposed by Pitones et al. ( 2019), holds for I(D Cn ), where C n denotes the cycle of length n. Miller generalized the concept of Alexander dual ideals of square-free monomial ideals to arbitrary monomial ideals, and in that direction, we study the Alexander dual of I(D G ) and its conditions to be Cohen-Macaulay.