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**The weak 3-flow conjecture and the weak circular flow conjecture.**
*(English)*
Zbl 1239.05083

Summary: We show that, for each natural number \(k>1\), every graph (possibly with multiple edges but with no loops) of edge-connectivity at least \(2k^{2}+k\) has an orientation with any prescribed outdegrees modulo \(k\) provided the prescribed outdegrees satisfy the obvious necessary conditions. For \(k=3\) the edge-connectivity 8 suffices. This implies the weak 3-flow conjecture proposed by F. Jaeger [Selected topics in graph theory, Vol. 3, 71- 95 (1988; Zbl 0658.05034)] (a natural weakening of Tutte’s 3-flow conjecture which is still open) and also a weakened version of the more general circular flow conjecture proposed by F. Jaeger [Colloq. Math. Soc. János Bolyai 37, No. 1, 391–402 (1984; Zbl 0567.05049)]. It also implies the tree-decomposition conjecture proposed in 2006 by Bárat and Thomassen when restricted to stars. Finally, it is the currently strongest partial result on the \((2+\epsilon )\)-flow conjecture by Goddyn and Seymour.

### MSC:

05C20 | Directed graphs (digraphs), tournaments |

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\textit{C. Thomassen}, J. Comb. Theory, Ser. B 102, No. 2, 521--529 (2012; Zbl 1239.05083)

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### References:

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