Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity

By Adam Van Tuyl. 2009

92 citations recorded by the source corpus.

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen–Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo–Mumford regularity of R/I(G) can be determined from the invariants of G.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0be4c82a3fea6d03bf41d4472a3182f435faa97e

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Citations in the corpus

Regularity of powers of bipartite graphs (2016, unknown): “In particular, there has been active research on bounding the homological invariants of edge ideals in terms of the combinatorial invariants of the associated graphs, see for example [3,10,12,15,17,19,22,23,26,28,32,33].”
Matchings, coverings, and Castelnuovo-Mumford regularity (2010, background): “(See his paper [33] for definitions and background.”
A Beginner’s Guide to Edge and Cover Ideals (2013, background): “Finally, if you are interested in learning more about the regularity of edge ideals, the following papers should be part of your reading list: [12, 32, 44, 46, 50, 62, 65, 66, 68].”
Algebraic study on Cameron-Walker graphs (2013, background): “, chordal graphs, unmixed bipartite graphs and sequentially Cohen–Macaulay bipartite graphs ([9, 12, 15]; see also [7, 13, 19, 20]).”
Edge ideals: algebraic and combinatorial properties (2010, background): “1, 4 [40] C. Francisco, H.T. Hà, and A. Van Tuyl, Associated primes of monomial ideals and odd holes in graphs, J. Algebraic Combin.”
On vertex decomposable simplicial complexes and their Alexander duals (2013, background): “More nice results on vertex decomposable simplicial complexes may be found in [1, 2, 5, 15, 17, 22, 24] and [25].”
Regularity of Squarefree Monomial Ideals (2013, background): “Then reg(G) = ν(G) + 1 in the following cases: (1) G is a sequentially Cohen-Macaulay bipartite graph (see [48]); (2) G is an unmixed bipartite graph (see [36]); (3) G is a very well-covered graph (see [38]); (4) G is a C5-free vertex decomposable graph (see [34], the case where G is also C4-free wa”