### Abstract

An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no K_{5}-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-K_{4}-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that ‘odd-K_{4}-minor-free’ cannot be replaced with ‘odd-K_{5}-minor-free.’ The main technical ingredient is a structural characterization of the class of odd-K_{4}-minor-free graphs, which is due to Lovász, Seymour, Schrijver, and Truemper.

Original language | English |
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Pages (from-to) | 1-14 |

Number of pages | 14 |

Journal | European Journal of Combinatorics |

Volume | 65 |

DOIs | |

Publication status | Published - 1 Oct 2017 |

Externally published | Yes |

### Cite this

_{4}-minor.

*European Journal of Combinatorics*,

*65*, 1-14. https://doi.org/10.1016/j.ejc.2017.04.010