Vertex decomposability and regularity of very well-covered graphs
Mahmoudi, Mohammad ; Mousivand, Amir ; Crupi, Marilena ; Rinaldo, Giancarlo ; Terai, Naoki ; Yassemi, Siamak
arXiv, 1006.1087 / Harvested from arXiv
Text on arXiv
A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that $G$ is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the edge ideal of $G$, we show that the Castelnuovo-Mumford regularity of $R/I(G)$ is equal to the maximum number of pairwise 3-disjoint edges of $G$. This improves Kummini's result on unmixed bipartite graphs.
Publié le : 2010-06-06
Classification:  Mathematics - Commutative Algebra,  Mathematics - Combinatorics,  13H10, 05C75 
@article{1006.1087,
     author = {Mahmoudi, Mohammad and Mousivand, Amir and Crupi, Marilena and Rinaldo, Giancarlo and Terai, Naoki and Yassemi, Siamak},
     title = {Vertex decomposability and regularity of very well-covered graphs},
     journal = {arXiv},
     volume = {2010},
     number = {0},
     year = {2010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1006.1087}
}

Mahmoudi, Mohammad; Mousivand, Amir; Crupi, Marilena; Rinaldo, Giancarlo; Terai, Naoki; Yassemi, Siamak. Vertex decomposability and regularity of very well-covered graphs. arXiv, Tome 2010 (2010) no. 0, . http://gdmltest.u-ga.fr/item/1006.1087/