### Abstract

The Eulerian recurrent length problem (ERLP) is defined as follows: Inputs to the ERLP consist of a positive integer k and a graph G. The objective of the ERLP is to determine whether there exists an Eulerian circuit, that is, a circuit that contains all of the edges of G, that has no subcycle of length less than k. If G has an Eulerian circuit that has no subcycle of length less than k, and every Eulerian circuit of G has an subcycle of length less than or equal to k, then k is called the Eulerian recurrent length of G, which is abbreviated to the ERL of G, and denoted by ERL(G). In this paper, the ERL’s of complete bipartite graphs are given. Furthermore, upper and lower bounds on the ERL’s of complete graphs are given. Let m and n be positive even integers with m ≥ n. It is shown that ERL(Km_n_ = 2n-4 if n = m ≥ 4, and ERL_Km_n_ = 2n otherwise, where Km_n is the complete bipartite graph whose vertex set V consists of two subsets A containing m vertices and B containing n vertices such that every edge of Km_n connects a vertex in A and one in B. It is also shown that n-4 ≤ ERL_Kn _ ≤ n-2 holds for every odd integer n greater than or equal to 7, where Kn is a complete graph consisting of n vertices.

Original language | English |
---|---|

Pages (from-to) | 2328-2333 |

Number of pages | 6 |

Journal | Advanced Science Letters |

Volume | 20 |

Issue number | 10-12 |

DOIs | |

Publication status | Published - 2014 |

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### Keywords

- Complete bipartite graph
- Complete graph
- Eulerian circuit
- Graph theory
- Hamiltonian decomposition
- Optimization problem

### ASJC Scopus subject areas

- Engineering(all)
- Environmental Science(all)
- Computer Science(all)
- Energy(all)
- Mathematics(all)
- Health(social science)
- Education

### Cite this

**The quest for the eulerian recurrent lengths of complete bipartite graphs and complete graphs.** / Jinbo, Shuji.

Research output: Contribution to journal › Article

*Advanced Science Letters*, vol. 20, no. 10-12, pp. 2328-2333. https://doi.org/10.1166/asl.2014.5723

}

TY - JOUR

T1 - The quest for the eulerian recurrent lengths of complete bipartite graphs and complete graphs

AU - Jinbo, Shuji

PY - 2014

Y1 - 2014

N2 - The Eulerian recurrent length problem (ERLP) is defined as follows: Inputs to the ERLP consist of a positive integer k and a graph G. The objective of the ERLP is to determine whether there exists an Eulerian circuit, that is, a circuit that contains all of the edges of G, that has no subcycle of length less than k. If G has an Eulerian circuit that has no subcycle of length less than k, and every Eulerian circuit of G has an subcycle of length less than or equal to k, then k is called the Eulerian recurrent length of G, which is abbreviated to the ERL of G, and denoted by ERL(G). In this paper, the ERL’s of complete bipartite graphs are given. Furthermore, upper and lower bounds on the ERL’s of complete graphs are given. Let m and n be positive even integers with m ≥ n. It is shown that ERL(Km_n_ = 2n-4 if n = m ≥ 4, and ERL_Km_n_ = 2n otherwise, where Km_n is the complete bipartite graph whose vertex set V consists of two subsets A containing m vertices and B containing n vertices such that every edge of Km_n connects a vertex in A and one in B. It is also shown that n-4 ≤ ERL_Kn _ ≤ n-2 holds for every odd integer n greater than or equal to 7, where Kn is a complete graph consisting of n vertices.

AB - The Eulerian recurrent length problem (ERLP) is defined as follows: Inputs to the ERLP consist of a positive integer k and a graph G. The objective of the ERLP is to determine whether there exists an Eulerian circuit, that is, a circuit that contains all of the edges of G, that has no subcycle of length less than k. If G has an Eulerian circuit that has no subcycle of length less than k, and every Eulerian circuit of G has an subcycle of length less than or equal to k, then k is called the Eulerian recurrent length of G, which is abbreviated to the ERL of G, and denoted by ERL(G). In this paper, the ERL’s of complete bipartite graphs are given. Furthermore, upper and lower bounds on the ERL’s of complete graphs are given. Let m and n be positive even integers with m ≥ n. It is shown that ERL(Km_n_ = 2n-4 if n = m ≥ 4, and ERL_Km_n_ = 2n otherwise, where Km_n is the complete bipartite graph whose vertex set V consists of two subsets A containing m vertices and B containing n vertices such that every edge of Km_n connects a vertex in A and one in B. It is also shown that n-4 ≤ ERL_Kn _ ≤ n-2 holds for every odd integer n greater than or equal to 7, where Kn is a complete graph consisting of n vertices.

KW - Complete bipartite graph

KW - Complete graph

KW - Eulerian circuit

KW - Graph theory

KW - Hamiltonian decomposition

KW - Optimization problem

UR - http://www.scopus.com/inward/record.url?scp=84919392897&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84919392897&partnerID=8YFLogxK

U2 - 10.1166/asl.2014.5723

DO - 10.1166/asl.2014.5723

M3 - Article

AN - SCOPUS:84919392897

VL - 20

SP - 2328

EP - 2333

JO - Advanced Science Letters

JF - Advanced Science Letters

SN - 1936-6612

IS - 10-12

ER -