On Conditional Connectivity of the Cartesian Product of Cycles

J.B. Saraf 1 , Y.M. Borse 2 , and Ganesh Mundhe 3
  • 1 Department of Mathematics, Amruteshwar Arts, Commerce and Science College, 412213, Vinzar, India
  • 2 Department of Mathematics, Savitribai Phule Pune University, 411007, Pune, India
  • 3 Army Institute of Technology, , 411015, Pune, India

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 HF 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 (2rh)arh, where arh is the number of vertices of a smallest h-regular subgraph of G.

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