On conditional connectivity of the Cartesian product of cycles

J. B. Saraf; Y. M. Borse; Ganesh Mundhe

On conditional connectivity of the Cartesian product of cycles

Combinatorics 2020-02-03 v1

Authors: J. B. Saraf , Y. M. Borse , Ganesh Mundhe

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 $r\geq 1$ cycles, each of length at least four and let $h$ be an integer such that $0\leq h\leq 2r-2$ . 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 $(2r-h)a_h^r$ , where $a_h^r$ is the number of vertices of a smallest $h$ -regular subgraph of $G.$

Keywords

conditional connectivity cartesian product cycles vertex connectivity edge connectivity regular subgraph

Cite

@article{arxiv.2001.11781,
  title  = {On conditional connectivity of the Cartesian product of cycles},
  author = {J. B. Saraf and Y. M. Borse and Ganesh Mundhe},
  journal= {arXiv preprint arXiv:2001.11781},
  year   = {2020}
}

On conditional connectivity of the Cartesian product of cycles

Abstract

Keywords

Cite

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