### Abstract

An Eulerian circuit of a graph is a circuit that contains all of the edges of the graph. A graph that has an Eulerian circuit is called an Eulerian graph. The Eulerian recurrent length of an Eulerian graph G is the maximum of the length of a shortest subcycle of an Eulerian circuit of G. In other words, if every Eulerian circuit of an Eulerian graph G has a subcycle of length less than or equal to l, and there is an Eulerian circuit of G that has no subcycle of length less than l, then the Eulerian recurrent length of G is l. The Eulerian recurrent length of graph G is abbreviated to the ERL of G, and denoted by ERL(G). In this paper, the ERL's of complete bipartite graphs are given. Let m and n be positive even integers with m ≥ n. It is shown that ERL(K _{m},n) 2n - 4 if n m ≥ 4, and ERL(K_{m},n) 2n otherwise. Furthermore, upper and lower bounds on the ERL's of complete graphs are given. It is shown that n - 4 ≤ ERL(K_{n}) ≤ n - 2 holds for every odd integer n greater than or equal to 7.

Original language | English |
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Title of host publication | IOP Conference Series: Materials Science and Engineering |

Publisher | Institute of Physics Publishing |

Volume | 58 |

Edition | 1 |

DOIs | |

Publication status | Published - 2014 |

Event | 2014 International Conference on Manufacturing, Optimization, Industrial and Material Engineering, MOIME 2014 - Jakarta, Indonesia Duration: Mar 29 2014 → Mar 30 2014 |

### Other

Other | 2014 International Conference on Manufacturing, Optimization, Industrial and Material Engineering, MOIME 2014 |
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Country | Indonesia |

City | Jakarta |

Period | 3/29/14 → 3/30/14 |

### Fingerprint

### ASJC Scopus subject areas

- Materials Science(all)
- Engineering(all)

### Cite this

*IOP Conference Series: Materials Science and Engineering*(1 ed., Vol. 58). [012019] Institute of Physics Publishing. https://doi.org/10.1088/1757-899X/58/1/012019

**On the Eulerian recurrent lengths of complete bipartite graphs and complete graphs.** / Jinbo, Shuji.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*IOP Conference Series: Materials Science and Engineering.*1 edn, vol. 58, 012019, Institute of Physics Publishing, 2014 International Conference on Manufacturing, Optimization, Industrial and Material Engineering, MOIME 2014, Jakarta, Indonesia, 3/29/14. https://doi.org/10.1088/1757-899X/58/1/012019

}

TY - GEN

T1 - On the Eulerian recurrent lengths of complete bipartite graphs and complete graphs

AU - Jinbo, Shuji

PY - 2014

Y1 - 2014

N2 - An Eulerian circuit of a graph is a circuit that contains all of the edges of the graph. A graph that has an Eulerian circuit is called an Eulerian graph. The Eulerian recurrent length of an Eulerian graph G is the maximum of the length of a shortest subcycle of an Eulerian circuit of G. In other words, if every Eulerian circuit of an Eulerian graph G has a subcycle of length less than or equal to l, and there is an Eulerian circuit of G that has no subcycle of length less than l, then the Eulerian recurrent length of G is l. The Eulerian recurrent length of graph G is abbreviated to the ERL of G, and denoted by ERL(G). In this paper, the ERL's of complete bipartite graphs are given. Let m and n be positive even integers with m ≥ n. It is shown that ERL(K m,n) 2n - 4 if n m ≥ 4, and ERL(Km,n) 2n otherwise. Furthermore, upper and lower bounds on the ERL's of complete graphs are given. It is shown that n - 4 ≤ ERL(Kn) ≤ n - 2 holds for every odd integer n greater than or equal to 7.

AB - An Eulerian circuit of a graph is a circuit that contains all of the edges of the graph. A graph that has an Eulerian circuit is called an Eulerian graph. The Eulerian recurrent length of an Eulerian graph G is the maximum of the length of a shortest subcycle of an Eulerian circuit of G. In other words, if every Eulerian circuit of an Eulerian graph G has a subcycle of length less than or equal to l, and there is an Eulerian circuit of G that has no subcycle of length less than l, then the Eulerian recurrent length of G is l. The Eulerian recurrent length of graph G is abbreviated to the ERL of G, and denoted by ERL(G). In this paper, the ERL's of complete bipartite graphs are given. Let m and n be positive even integers with m ≥ n. It is shown that ERL(K m,n) 2n - 4 if n m ≥ 4, and ERL(Km,n) 2n otherwise. Furthermore, upper and lower bounds on the ERL's of complete graphs are given. It is shown that n - 4 ≤ ERL(Kn) ≤ n - 2 holds for every odd integer n greater than or equal to 7.

UR - http://www.scopus.com/inward/record.url?scp=84906542905&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906542905&partnerID=8YFLogxK

U2 - 10.1088/1757-899X/58/1/012019

DO - 10.1088/1757-899X/58/1/012019

M3 - Conference contribution

AN - SCOPUS:84906542905

VL - 58

BT - IOP Conference Series: Materials Science and Engineering

PB - Institute of Physics Publishing

ER -