## Abstract

Extending an old conjecture of Tutte, Jaeger conjectured in 1988 that for any fixed integer p ≥ 1, the edges of any 4p-edge connected graph can be oriented so that the difference between the outdegree and the indegree of each vertex is divisible by 2p+1. It is known that it suffices to prove this conjecture for (4p+1)-regular, 4p-edge connected graphs. Here we show that there exists a finite p0 such that for every p > p0 the assertion of the conjecture holds for all (4p+1)-regular graphs that satisfy some mild quasi-random properties, namely, the absolute value of each of their non-trivial eigenvalues is at most c1p^{2/3} and the neighbourhood of each vertex contains at most c2p^{3/2} edges, where c1, c2 > 0 are two absolute constants. In particular, this implies that for p > p0 the assertion of the conjecture holds asymptotically almost surely for random (4p+1)-regular graphs.

Original language | English (US) |
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Pages (from-to) | 321-329 |

Number of pages | 9 |

Journal | Combinatorics Probability and Computing |

Volume | 20 |

Issue number | 3 |

DOIs | |

State | Published - May 2011 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics