Importance: High ✭✭✭
Author(s): Bouchet, Andre
Recomm. for undergrads: no
Prize: none
Posted by: mdevos
on: March 7th, 2007
Conjecture   Every bidirected graph with a nowhere-zero $ k $-flow for some $ k $, has a nowhere-zero $ 6 $-flow.

Definition: A bidirected graph is a graph in which every edge has two arrowheads, one next to each endpoint. If the edge $ e $ has ends $ u $ and $ v $, then the arrowheads nearest $ u $ and $ v $ may point either toward $ u $ or toward $ v $ (giving four possibilities in all). If $ G $ is a bidirected graph, a $ k $-flow of G is a map $ \phi:E(G)\to \{-(k-1),...,-1,0,1,...,k-1\} $ with the property that at every vertex, the sum of $ \phi $ on the edges whose ends at $ v $ are directed into $ v $ is equal to the sum of $ \phi $ on the edges whose ends at $ v $ are directed out of $ v $. We say that $ \phi $ is nowhere-zero if $ \phi(e) \neq 0 $ for every $ e \in E(G) $ (see nowhere-zero flows).

A bidirected Orientation of the Petersen graph

Flows on bidirected graphs arise naturally as duals of local-tensions on a non-orientable surface. For more on this relationship, see [B]. Bouchet proved that the above conjecture is true with 6 replaced by 216, and exhibited a bidirected Petersen graph as above which shows that 6 is the best value possible. Zyka [Z] and independently Fouquet improved upon this result proving that the above conjecture is true with 6 replaced by 30. Khelladi [K] proved that for 4-connected graphs, the above conjecture is true with 6 replaced by 18. DeVos [D] proved that the above conjecture holds with 6 replaced by 12, and showed that every 4-edge-connected bidirected graph with a nowhere-zero integer flow also has a nowhere-zero 4-flow.

Bibliography

[B] A. Bouchet, Nowhere-Zero Integral Flows on a Bidirected Graph, J. Combinatorial Theory Ser. B 34 (1983) 279-292. MathSciNet

[D] M. DeVos, Flows on Bidirected Graphs, preprint.

[K] A. Khelladi, Nowhere-Zero Integral Chains and Flows in Bidirected Graphs, J. Combinatorial Theory Ser. B 43 (1987) 95-115. MathSciNet

[Z] O. Zyka, Bidirected Nowhere-Zero Flows, Thesis, Charles University, Praha (1988).


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