Let \(I\) be a square-free monomial ideal, and let \(H_I\) denote its associated hypergraph. We introduce the notion of generalized \(k\)-admissible matchings for hypergraphs, extending the concept of k-admissible matchings for graphs due to Erey and Hibi. Using this framework, we establish a sharp lower bound for the Castelnuovo-Mumford regularity of the \(k\)-th square-free power \(I^{[k]}\).
In the case when \(I\) is equigenerated in degree \(d\), this lower bound can be described using a combinatorial invariant \(aim(H_I,k)\), called the \(k\)-admissible matching number of \(H_I\). More specifically, we prove that \(reg(I^{[k]}) \geq (d-1) aim(H_I,k)+k\) whenever \(I^{[k]}\neq 0\).
Even in the special case of edge ideals \(I(G)\), this yields the first general lower bound for \(reg(I(G)^{[k]})\) in terms of \(k\)-admissible matchings. Moreover, when \(G\) is a forest, the invariant \(aim(G,k)\) coincides with the classical \(k\)-admissible matching number.
We further conjecture that this lower bound is attained for chordal graphs. In support of this conjecture, we present two cases in which the general lower bound is achieved. We show that if \(G\) is a block graph, then \(reg(I(G)^{[k]})= aim(G,k)+k\). Additionally, for a Cohen-Macaulay chordal graph \(G\), we show that \(reg(I(G)^{[2]})= aim(G,2)+2\).
This talk is based on a recent work with A. Roy and K. Saha.
