Abstract
We describe a combinatorial condition on a graphwhich guarantees that all powers of its vertex cover ideal are componentwise linear. Then motivated by Eagon and Reiner’s Theorem we study whether all powers of the vertex cover ideal of a Cohen-Macaulay graph have linear free resolutions. After giving a complete characterization of Cohen-Macaulay cactus graphs (i.e., connected graphs in which each edge belongs to at most one cycle) we show that all powers of their vertex cover ideals have linear resolutions.
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