The weak 3-flow conjecture and the weak circular flow conjecture

Authors:
Carsten Thomassen
Affiliations:
Department of Mathematics, Technical University of Denmark, DK-2800 Lyngby, Denmark and King Abdulaziz University, Jeddah, Saudi Arabia
Venue:
Journal of Combinatorial Theory Series B
Year:
2012

Citing 4
Cited 2

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Nowhere-zero 3-flows and modulo k-orientations

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Abstract

We show that, for each natural number k1, 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 in 1988 by Jaeger (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 Jaeger in 1982. It also implies the tree-decomposition conjecture proposed in 2006 by Barat and Thomassen when restricted to stars. Finally, it is the currently strongest partial result on the (2+@e)-flow conjecture by Goddyn and Seymour.