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Register a new userCohen-Macaulay edge-weighted edge ideals of very well-covered graphs
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We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.
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Let C be a uniform clutter and let I=I(C) be its edge ideal. We prove that if C satisfies the packing property (resp. max-flow min-cut property), then there is a uniform Cohen-Macaulay clutter C1 satisfying the packing property (resp. max-flow min-cut property) such that C is a minor of C1. For arbitrary edge ideals of clutters we prove that the normality property is closed under parallelizations. Then we show some applications to edge ideals and clutters which are related to a conjecture of Conforti and Cornuejols and to max-flow min-cut problems.
Let $D$ be a weighted oriented graph, whose underlying graph is $G$, and let $I(D)$ be its edge ideal. If $G$ has no $3$-, $5$-, or $7$-cycles, or $G$ is K{o}nig, we characterize when $I(D)$ is unmixed. If $G$ has no $3$- or $5$-cycles, or $G$ is Konig, we characterize when $I(D)$ is Cohen--Macaulay. We prove that $I(D)$ is unmixed if and only if $I(D)$ is Cohen--Macaulay when $G$ has girth greater than $7$ or $G$ is Konig and has no $4$-cycles.
Let $mathcal{D}$ be a weighted oriented graph and $I(mathcal{D})$ be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of $I(mathcal{D})$ coincide when $mathcal{D}$ is a weighted oriented certain class of tree. Finally, we give necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented lines.
Scattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen-Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial ideals. The purpose of this survey is to gather the developments into one location, with self-contained proofs, including direct combinatorial topological connections between them.
Let $G=(V,E)$ be a graph. If $G$ is a Konig graph or $G$ is a graph without 3-cycles and 5-cycle, we prove that the following conditions are equivalent: $Delta_{G}$ is pure shellable, $R/I_{Delta}$ is Cohen-Macaulay, $G$ is unmixed vertex decomposable graph and $G$ is well-covered with a perfect matching of Konig type $e_{1},...,e_{g}$ without square with two $e_i$s. We characterize well-covered graphs without 3-cycles, 5-cycles and 7-cycles. Also, we study when graphs without 3-cycles and 5-cycles are vertex decomposable or shellable. Furthermore, we give some properties and relations between critical, extendables and shedding vertices. Finally, we characterize unicyclic graphs with each one of the following properties: unmixed, vertex decomposable, shellable and Cohen-Macaulay.
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