For every bipartite graph G and every s >= 2, the Castelnuovo-Mumford regularity of the s-th power of its edge ideal is at most 2s + reg(I(G)) - 2, and this bound is tight.
Upper bounds for the regularity of powers of edge ideals of graphs
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abstract
Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of $I(G)^q$ in terms of certain combinatorial invariants associated with $G$. We also prove a weaker version of a conjecture by Alilooee, Banerjee, Beyarslan and H\`a on an upper bound for the regularity of $I(G)^q$ and we prove the conjectured upper bound for the class of vertex decomposable graphs. Using these results, we explicitly compute the regularity of $I(G)^q$ for several classes of graphs.
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math.AC 1years
2019 1verdicts
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Regularity of Edge Ideals Via Suspension
For every bipartite graph G and every s >= 2, the Castelnuovo-Mumford regularity of the s-th power of its edge ideal is at most 2s + reg(I(G)) - 2, and this bound is tight.