A Characterization of Graphs Whose Small Powers of Their Edge Ideals Have a Linear Free Resolution

Minh, Nguyen Cong; Vu, Thanh

doi:10.1007/s00493-023-00074-z

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2024

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“…Finally we conclude by a discussion on a related conjecture by [27]: Very recently this conjecture was verified for s = 2 and s = 3 by Minh and Vu [24]. Theorem 2.3 was proved by Banerjee in his thesis to study this conjecture and related other problems, based on Theorem 2.2.…”

Section: Further Researchmentioning

confidence: 64%

Regularity of Edge Ideals Via Suspension

Banerjee,

Nevo

2024

Algebraic Combinatorics

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We study the Castelnuovo-Mumford regularity of powers of edge ideals for arbitrary (finite simple) graphs. It has been repeatedly conjectured that for every graph G, reg(I(G) s ) ⩽ 2s + reg I(G) − 2 for all s ⩾ 2, which is the best possible upper bound for any s. We prove this conjecture for every s for all bipartite graphs, and for s = 2 for all graphs. The s = 2 case is crucial for our work and suspension plays a key role in its proof.

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Section: Further Researchmentioning

confidence: 64%