### Abstract

Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V, E) having a k-edge-connected T-odd orientation for every subset T ⊆ V with |E| + |T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k)-edge-connected graph with |V| + |E| even has a (k - 1)-edge-connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k edge-disjoint paths from s to every node and k-1 edge-disjoint paths from every node to s.

Original language | English |
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Pages (from-to) | 47-70 |

Number of pages | 24 |

Journal | Combinatorica |

Volume | 22 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 7 2002 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*22*(1), 47-70. https://doi.org/10.1007/s004930200003