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.
Regularity of powers of edge ideals: from local properties to global bounds
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abstract
Let $I = I(G)$ be the edge ideal of a graph $G$. We give various general upper bounds for the regularity function $\text{reg} I^s$, for $s \ge 1$, addressing a conjecture made by the authors and Alilooee. When $G$ is a gap-free graph and locally of regularity 2, we show that $\text{reg} I^s = 2s$ for all $s \ge 2$. This is a slightly weaker version of a conjecture of Nevo and Peeva. Our method is to investigate the regularity function $\text{reg}I^s$, for $s \ge 1$, via local information of $I$.
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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.