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## Abstract

The conditional *h*-vertex (*h*-edge) connectivity of a connected graph *H* of minimum degree *k > h* is the size of a smallest vertex (edge) set *F* of *H* such that *H* − *F* is a disconnected graph of minimum degree at least *h.* Let *G* be the Cartesian product of *r* ≥ 1 cycles, each of length at least four and let *h* be an integer such that 0 ≤ *h* ≤ 2*r* − 2. In this paper, we determine the conditional *h*-vertex-connectivity and the conditional *h*-edge-connectivity of the graph *G.* We prove that both these connectivities are equal to (2*r*−*h*)*a ^{r}_{h}*, where

*a*is the number of vertices of a smallest

^{r}_{h}*h*-regular subgraph of

*G.*