### Abstract

The concepts of k-pairable graphs and the pair length of a graph were introduced by Chen [Discrete Math. 287 (2004), 11-15] to generalize an elegant result of Graham et al. [Amer. Math. Monthly 101 (1994), 664- 667] from hypercubes and graphs with antipodal isomorphisms to a much larger class of graphs. A graph G is k-pairable if there is a positive integer k such that 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 a graph G, denoted by p(G), is the maximum positive integer k such that G is k-pairable; and p(G) = 0 if G is not k-pairable for any positive integer k. The aim of this paper is to answer an open question posted in our previous paper [Discrete Math. 310 (2010), 3334-3350]; that is, the question of determining the minimum order of a graph in the set of r-regular bipartite graphs of pair length k. We solve the problem for all positive integers k and r except for the case when both k ≥ 5 and r ≥ 3 are odd. For the case that is still open, we provide bounds on the minimum order concerned. Also we post a conjecture on the minimum order of a cubic bipartite graph of pair length k for any odd number k > 1.

Original language | English (US) |
---|---|

Pages (from-to) | 50-65 |

Number of pages | 16 |

Journal | Australasian Journal of Combinatorics |

Volume | 66 |

Issue number | 1 |

State | Published - Jan 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*66*(1), 50-65.

}

*Australasian Journal of Combinatorics*, vol. 66, no. 1, pp. 50-65.

**Minimum order of r-regular bipartite graphs of pair length k.** / Che, Zhongyuan; Chen, Zhibo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Minimum order of r-regular bipartite graphs of pair length k

AU - Che, Zhongyuan

AU - Chen, Zhibo

PY - 2016/1/1

Y1 - 2016/1/1

N2 - The concepts of k-pairable graphs and the pair length of a graph were introduced by Chen [Discrete Math. 287 (2004), 11-15] to generalize an elegant result of Graham et al. [Amer. Math. Monthly 101 (1994), 664- 667] from hypercubes and graphs with antipodal isomorphisms to a much larger class of graphs. A graph G is k-pairable if there is a positive integer k such that 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 a graph G, denoted by p(G), is the maximum positive integer k such that G is k-pairable; and p(G) = 0 if G is not k-pairable for any positive integer k. The aim of this paper is to answer an open question posted in our previous paper [Discrete Math. 310 (2010), 3334-3350]; that is, the question of determining the minimum order of a graph in the set of r-regular bipartite graphs of pair length k. We solve the problem for all positive integers k and r except for the case when both k ≥ 5 and r ≥ 3 are odd. For the case that is still open, we provide bounds on the minimum order concerned. Also we post a conjecture on the minimum order of a cubic bipartite graph of pair length k for any odd number k > 1.

AB - The concepts of k-pairable graphs and the pair length of a graph were introduced by Chen [Discrete Math. 287 (2004), 11-15] to generalize an elegant result of Graham et al. [Amer. Math. Monthly 101 (1994), 664- 667] from hypercubes and graphs with antipodal isomorphisms to a much larger class of graphs. A graph G is k-pairable if there is a positive integer k such that 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 a graph G, denoted by p(G), is the maximum positive integer k such that G is k-pairable; and p(G) = 0 if G is not k-pairable for any positive integer k. The aim of this paper is to answer an open question posted in our previous paper [Discrete Math. 310 (2010), 3334-3350]; that is, the question of determining the minimum order of a graph in the set of r-regular bipartite graphs of pair length k. We solve the problem for all positive integers k and r except for the case when both k ≥ 5 and r ≥ 3 are odd. For the case that is still open, we provide bounds on the minimum order concerned. Also we post a conjecture on the minimum order of a cubic bipartite graph of pair length k for any odd number k > 1.

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UR - http://www.scopus.com/inward/citedby.url?scp=84983413400&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84983413400

VL - 66

SP - 50

EP - 65

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

IS - 1

ER -