×

Cohen-Macaulay bipartite graphs. (English) Zbl 0869.13003

Let \(G\) be a graph on the vertex set \(V=\{x_1, \dots, x_n\}\). Let \(k\) be a field and let \(R\) be the polynomial ring \(k[x_1, \dots, x_n]\). The graph ideal \(I(G)\), associated to \(G\), is the ideal of \(R\) generated by the set of square-free monomials \(x_ix_j\) so that \(x_i\) is adjacent to \(x_j\). The graph \(G\) is Cohen-Macaulay over \(k\) if \(R/I(G)\) is a Cohen-Macaulay ring. Let \(G\) be a Cohen-Macaulay bipartite graph. The main result of this paper shows that \(G \backslash \{v\}\) is Cohen-Macaulay for some vertex \(v\) in \(G\). Then as a consequence it is shown that the Stanley-Reisner simplicial complex of \(I(G)\) is shellable. An example of N. Terai is presented showing these results fail for Cohen-Macaulay non-bipartite graphs.

MSC:

13C14 Cohen-Macaulay modules
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
05C90 Applications of graph theory
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
55U05 Abstract complexes in algebraic topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] W. Bruns and J. Herzog, Cohen-Macaulay Rings. Cambridge 1993. · Zbl 0788.13005
[2] Bruns, W.; Hibi, T., Stanley-Reisner rings with pure resolutions, Comm. Algebra, 21, 1201-1217 (1995) · Zbl 0823.13015 · doi:10.1080/00927879508825274
[3] Cavaliere, M.; Rossi, M.; Valla, G., On short graded algebras (Proc. Workshop in Commutative Algebra, Salvador, Brazil), LNM, 1430, 21-31 (1990) · Zbl 0714.13012
[4] R. Fröberg, On Stanley-Reisner rings. In: Topics in algebra, Part 2, S. Balcerzyk et. al., eds., 57-70, 1990. · Zbl 0741.13006
[5] F. Harary, Graph Theory. Reading, MA, 1972. · Zbl 0235.05105
[6] Reisner, G., Cohen-Macaulay quotients of polynomial rings, Adv. in Math., 21, 31-49 (1976) · Zbl 0345.13017 · doi:10.1016/0001-8708(76)90114-6
[7] Simis, A.; Vasconcelos, W. V.; Villarreal, R., On the ideal theory of graphs, J. Algebra, 167, 389-416 (1994) · Zbl 0816.13003 · doi:10.1006/jabr.1994.1192
[8] R. Stanley, Combinatorics and Commutative Algebra. Basel-Boston-Berlin 1983. · Zbl 0537.13009
[9] Villarreal, R., Cohen-Macaulay graphs, Manuscripta Math., 66, 277-293 (1990) · Zbl 0737.13003 · doi:10.1007/BF02568497
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.