3,928 research outputs found
Cohen-Macaulay binomial edge ideals
We study the depth of classes of binomial edge ideals and classify all closed
graphs whose binomial edge ideal is Cohen--Macaulay.Comment: 9 page
Binomial edge ideals of bipartite graphs
Binomial edge ideals are a noteworthy class of binomial ideals that can be associated with graphs, generalizing the ideals of 2-minors. For bipartite graphs we prove the converse of Hartshorne’s Connectedness Theorem, according to which if an ideal is Cohen–Macaulay, then its dual graph is connected. This allows us to classify Cohen–Macaulay binomial edge ideals of bipartite graphs, giving an explicit and recursive construction in graph-theoretical terms. This result represents a binomial analogue of the celebrated characterization of (monomial) edge ideals of bipartite graphs due to Herzog and Hibi (2005). Herzog J., Hibi T. Distributive lattices, bipartite graphs and Alexander duality J. Algebraic Combin., 22 (2005), pp. 289-30
Generalized Binomial Edge Ideals
This paper studies a class of binomial ideals associated to graphs with
finite vertex sets. They generalize the binomial edge ideals, and they arise in
the study of conditional independence ideals. A Gr\"obner basis can be computed
by studying paths in the graph. Since these Gr\"obner bases are square-free,
generalized binomial edge ideals are radical. To find the primary decomposition
a combinatorial problem involving the connected components of subgraphs has to
be solved. The irreducible components of the solution variety are all rational.Comment: 6 pages. arXiv admin note: substantial text overlap with
arXiv:1110.133
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