### Abstract

In this paper we consider a graph G, a partition P = {V_{1}, V_{2},: :: , V_{κ}} of V (G) and the generalized complements G^{P} _{κ} and G^{P} _{κ(i)} with respect to the partition P. We derive conditions to be satisfied by P so that G commutes with its generalized complements. Apart from the general characterization, we also obtain conditions on P = {V_{1}, V_{2},: :: , V_{κ}} so that G commutes with its generalized complements for certain classes of graphs namely complete graphs, cycles and generalized wheels. In the process we obtain a commuting decomposition of regular complete k-partite graph K_{n1,n2,:::,nk} in terms of a Hamiltonian cycle and its kcomplement. We also get a commuting decomposition of a complete k-partite graph K_{n1,n2,:::,nk} in terms of a generalized wheel and its κ- complement, where _{n1,n2,:::,nk} satisfy some conditions.

Original language | English |
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Pages (from-to) | 63-84 |

Number of pages | 22 |

Journal | Malaysian Journal of Mathematical Sciences |

Volume | 12 |

Issue number | 1 |

Publication status | Published - 01-01-2018 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Malaysian Journal of Mathematical Sciences*,

*12*(1), 63-84.