### Abstract

The k-pairable graphs, introduced by Chen in 2004, constitute a wide class of graphs with a new type of symmetry, which includes many graphs of theoretical and practical importance, such as hypercubes, Hamming graphs of even order, antipodal graphs (also called diametrical graphs, or symmetrically even graphs), S-graphs, etc. Let k be a positive integer. A connected graph G is said to be k-pairable if the automorphism group of G contains an involution φ with the property that the distance between x and φ(x) is at least k for any vertex x of G. The pair length of G is k if G is k-pairable but not (k + 1)-pairable. It is known that any graph of pair length k > 0 has even order at least 2k. In this paper, we give sharp bounds for the size of a graph G of order n and pair length k for any integer k > 0 and any even integer n ≥ 2k, when G is bipartite and when G is not restricted to be bipartite, respectively.

Original language | English (US) |
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Pages (from-to) | 172-183 |

Number of pages | 12 |

Journal | Australasian Journal of Combinatorics |

Volume | 62 |

Issue number | 2 |

State | Published - Jul 24 2015 |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

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## Cite this

*Australasian Journal of Combinatorics*,

*62*(2), 172-183.