‎Unmixed bipartite graphs have been characterized by Ravindra and‎ ‎Villarreal independently‎. ‎Our aim in this paper is to‎ ‎characterize unmixed $r$-partite graphs under a certain condition‎, ‎which is a generalization of Villarreal's theorem on bipartite‎ ‎graphs‎. ‎Also, we give some examples and counterexamples in relevance to this subject‎.

The correspondence between unmixed bipartite graphs and sublattices of the Boolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.

Weak Bourbaki unmixed rings are defined in this paper. The definition of a weak Bourbaki unmixed ring is a candidate for an “appropriate” definition of CohenMacaulayness. We will see that this definition satisfies many of the conditions we want an “appropriate” definition to satisfy. It is not yet known whether this definition (or any other) satisfies all of the conditions. However, no example ...

A ΣΠΣΠ(k) circuit C = ∑k i=1 Fi = ∑k i=1 ∏di j=1 fij is unmixed if for each i ∈ [k], Fi = fi1(x1) · · · fin(xn), where each fij is a univariate polynomial given in the sparse representation. In this paper, we give a polynomial time black-box algorithm of identity testing for the low degree unmixed ΣΠΣΠ(k) circuits. In order to obtain the black-box algorithm, we first show that a special class o...

For an unmixed bipartite graph G we consider the lattice of vertex covers LG and compute depth, projective dimension and extremal Bettinumbers of R/I(G) in terms of this lattice.