Flows And Jaeger Conjecture
Attention For the purpose of this project we modify the notion of resigning and apply a dual notion as follows. It is also important to consider multigraphs rather than just simple graphs for this project.
Given a signed graph (G, 𝛴 ) a cycle-resigning is to multiply signs of all edges of a cycle by a - sign. When G is planar, this corresponds to the dual of the vertex resigning we normally work with.
Given a signed graph (G, 𝛴 ) and an orientation of G, we associate to each vertex v of G two values of f+(v) and f-(v) as follows: f+(v) is the difference between incoming minus outgoing value at vertex v among positive edges incident to v and f-(v) is the difference between incoming minus outgoing value at vertex v among negative edges at this vertex.
The main conjecture here is that: given a positive integer k, under a main condition of multigraph G being Eulerian, a necessary connectivity of high connectivity (as a function of k) and (perhaps unnecessary) condition of planarity, for any choice of signature 𝛴 we may always find a cycle-resigning 𝛴' and an orientation of G such that f+(v) and f-(v) with respect to 𝛴' and the orientation satisfies the following:
f+(v)=(2k-1) f-(v)
What is true is that when connectivity is high enough this is always doable (higher than 8k should follow from a work of Charpentier, Naseraser, Sopena-in preparation). The main question is to find the best bound on connectivity that works. With condition of planarity as analogue of Jaeger-Zhang conjecture one may expect a connectivity of 4k-2 to be enough.
The roll of planarity is yet to be examined. Perhaps a similar result would holds for general graphs or perhaps we have to do some adjustment to the equation f+(v)=(2k-1) f-(v). In light of recent disproof of Jager's conjecture, once a correct analogue is formulated, we expect connectivity of 4k-2 not be enough but connectivity of 6k to be enough, thus expecting the best bound to be somewhere between 4k and 6k..