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In this paper, we study the s-th symbolic powers of the edge ideals of complete graphs. In particular, we provide a criterion for finding an Eliahou–Kervaire splitting on these ideals, and we use the splitting to provide a description for the graded Betti numbers. We also discuss the symbolic powers and graded Betti numbers of edge ideals of parallelizations of finite simple graphs.
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Cooper et al. (2024) studied this question.
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