On conditional connectivity of the Cartesian product of cycles

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 ≤ 2r − 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 (2r − h)a r h , where a r h is the number of vertices of a smallest h-regular subgraph of G.