### Abstract

If the degree of every vertex of a connected graph is even, then the graph has a circuit that contains all of edges, namely an Eulerian circuit. If the length of a shortest subcycle of an Eulerian circuit of a given graph is the largest, then the length is called the Eulerian recurrent length of the graph. For an odd integer n greater than or equal to 3, e(n) denotes the Eulerian recurrent length of the complete graph with n vertices. Values e(n) for all odd integers n with have been found by verification experiments using computers. If n is 7, 9, 11, or 13, then holds, for example. On the other hand, it has been shown that holds for any odd integer n greater than or equal to 15 in previous researches. In this paper, it is proved that holds for every odd integer n greater than or equal to 15. In the core part of the proof of the main theorem, an IP (integer programming) solver is used as the amount of computation is too large to be solved by hand.

Original language | English |
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Title of host publication | WALCOM |

Subtitle of host publication | Algorithms and Computation - 13th International Conference, WALCOM 2019, Proceedings |

Editors | Shin-ichi Nakano, Gautam K. Das, Partha S. Mandal, Krishnendu Mukhopadhyaya |

Publisher | Springer Verlag |

Pages | 199-208 |

Number of pages | 10 |

ISBN (Print) | 9783030105631 |

DOIs | |

Publication status | Published - Jan 1 2019 |

Event | 13th International Conference and Workshop on Algorithms and Computations, WALCOM 2019 - Guwahati, India Duration: Feb 27 2019 → Mar 2 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11355 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 13th International Conference and Workshop on Algorithms and Computations, WALCOM 2019 |
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Country | India |

City | Guwahati |

Period | 2/27/19 → 3/2/19 |

### Fingerprint

### Keywords

- Complete graphs
- Computer experiments
- Eulerian circuits
- Graph theory
- Shortest subcycles

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*WALCOM: Algorithms and Computation - 13th International Conference, WALCOM 2019, Proceedings*(pp. 199-208). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11355 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-10564-8_16

**The Upper Bound on the Eulerian Recurrent Lengths of Complete Graphs Obtained by an IP Solver.** / Jinbo, Shuji; Maruoka, Akira.

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

*WALCOM: Algorithms and Computation - 13th International Conference, WALCOM 2019, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11355 LNCS, Springer Verlag, pp. 199-208, 13th International Conference and Workshop on Algorithms and Computations, WALCOM 2019, Guwahati, India, 2/27/19. https://doi.org/10.1007/978-3-030-10564-8_16

}

TY - GEN

T1 - The Upper Bound on the Eulerian Recurrent Lengths of Complete Graphs Obtained by an IP Solver

AU - Jinbo, Shuji

AU - Maruoka, Akira

PY - 2019/1/1

Y1 - 2019/1/1

N2 - If the degree of every vertex of a connected graph is even, then the graph has a circuit that contains all of edges, namely an Eulerian circuit. If the length of a shortest subcycle of an Eulerian circuit of a given graph is the largest, then the length is called the Eulerian recurrent length of the graph. For an odd integer n greater than or equal to 3, e(n) denotes the Eulerian recurrent length of the complete graph with n vertices. Values e(n) for all odd integers n with have been found by verification experiments using computers. If n is 7, 9, 11, or 13, then holds, for example. On the other hand, it has been shown that holds for any odd integer n greater than or equal to 15 in previous researches. In this paper, it is proved that holds for every odd integer n greater than or equal to 15. In the core part of the proof of the main theorem, an IP (integer programming) solver is used as the amount of computation is too large to be solved by hand.

AB - If the degree of every vertex of a connected graph is even, then the graph has a circuit that contains all of edges, namely an Eulerian circuit. If the length of a shortest subcycle of an Eulerian circuit of a given graph is the largest, then the length is called the Eulerian recurrent length of the graph. For an odd integer n greater than or equal to 3, e(n) denotes the Eulerian recurrent length of the complete graph with n vertices. Values e(n) for all odd integers n with have been found by verification experiments using computers. If n is 7, 9, 11, or 13, then holds, for example. On the other hand, it has been shown that holds for any odd integer n greater than or equal to 15 in previous researches. In this paper, it is proved that holds for every odd integer n greater than or equal to 15. In the core part of the proof of the main theorem, an IP (integer programming) solver is used as the amount of computation is too large to be solved by hand.

KW - Complete graphs

KW - Computer experiments

KW - Eulerian circuits

KW - Graph theory

KW - Shortest subcycles

UR - http://www.scopus.com/inward/record.url?scp=85062675847&partnerID=8YFLogxK

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U2 - 10.1007/978-3-030-10564-8_16

DO - 10.1007/978-3-030-10564-8_16

M3 - Conference contribution

AN - SCOPUS:85062675847

SN - 9783030105631

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 199

EP - 208

BT - WALCOM

A2 - Nakano, Shin-ichi

A2 - Das, Gautam K.

A2 - Mandal, Partha S.

A2 - Mukhopadhyaya, Krishnendu

PB - Springer Verlag

ER -