Jaeger's modular orientation conjecture

Importance: High ✭✭✭
Author(s): Jaeger, Francois
Recomm. for undergrads: no
Prize: none
Posted by: mdevos
on: March 7th, 2007
Conjecture   Every $ 4k $-edge-connected graph can be oriented so that $ {\mathit indegree}(v) - {\mathit outdegree}(v) \cong 0 $ (mod $ 2k+1 $) for every vertex $ v $.

Jaeger called an orientation with the above property a modular $ (2k+1) $-orientation, and observed that a graph has a modular $ (2k+1) $-orientation if and only if it has a $ (2+\frac{1}{k}) $-flow. Thus, this conjecture may be seen as a sharp form of the 2+epsilon flow conjecture. For k=1, this problem is precisely the 3-flow conjecture, and for k=2, Jaeger showed that this conjecture (if true) would imply the 5-flow conjecture. If true, this conjecture would be best possible for every value of k.

The restriction of this conjecture to planar graphs is open, and has a dual formulation. See Mapping planar graphs to odd cycles.


[J] F. Jaeger, On circular flows in graphs. Finite and infinite sets, Vol. I, II (Eger, 1981), 391--402, Colloq. Math. Soc. János Bolyai, 37, North-Holland, Amsterdam, 1984.. MathSciNet

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