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Binomial edge ideals of bipartite graphs. (English) Zbl 1384.05094

Summary: 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 J. Herzog and T. Hibi [J. Algebr. Comb. 22, No. 3, 289–302 (2005; Zbl 1090.13017)].

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

Citations:

Zbl 1090.13017

Software:

Macaulay2; nauty; Traces
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Full Text: DOI arXiv

References:

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