### Abstract

The concept of a k-pairable graph was introduced by Z. Chen [On k-pairable graphs, Discrete Mathematics 287 (2004), 11-15] as an extension of hypercubes and graphs with an antipodal isomorphism. In the present paper we generalize further this concept of a k-pairable graph to the concept of a semi-pairable graph. We prove that a graph is semi-pairable if and only if its prime factor decomposition contains a semi-pairable prime factor or some repeated prime factors. We also introduce a special class of k-pairable graphs which are called uniquely k-pairable graphs. We show that a graph is uniquely pairable if and only if its prime factor decomposition has at least one pairable prime factor, each prime factor is either uniquely pairable or not semi-pairable, and all prime factors which are not semi-pairable are pairwise non-isomorphic. As a corollary we give a characterization of uniquely pairable Cartesian product graphs.

Original language | English (US) |
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Pages (from-to) | 6104-6110 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 308 |

Issue number | 24 |

DOIs | |

Publication status | Published - Dec 28 2008 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*308*(24), 6104-6110. https://doi.org/10.1016/j.disc.2007.11.029