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For a finite abelian group $G$ and a positive integer $h$, the unrestricted(resp.~restricted) $h$-critical number $\chi(G,h)$ (resp.~$\chi \hat{\;}(G,h)$)of $G$ is defined to be the minimum value of $m$, if exists, for which the$h$-fold unrestricted (resp.~restricted) sumset of every $m$-subset of $G$equals $G$ itself. Here we determine $\chi(G,h)$ for all $G$ and $h$; and proveseveral results for $\chi \hat{\;}(G,h)$, including the cases of any $G$ and $h= 2$, any $G$ and large $h$, and any $h$ for the cyclic group $\mathbb{Z}_n$ ofeven order. We also provide a lower bound for $\chi \hat{\;}(\mathbb{Z}_n,3)$that we believe is exact for every $n$---this conjecture is a generalization ofthe one made by Gallardo, Grekos, et al.~that was proved (for large $n$) byLev.

The $h$-critical number of finite abelian groups