Bouchet's 6-flow conjecture

Bouchet's 6-flow conjecture

Definitions and mathematical statement

A finite bidirected multigraph consists of an undirected multigraph together with a sign on each incidence, or half-edge, at a vertex. Use sign $+1$ for an end pointing into its vertex and $-1$ for an end pointing out. The two ends of an edge are chosen independently: both may point in, both out, or in opposite directions. A loop has two incidences at the same vertex, each counted separately in the sums below.

For an integer $q\geq2$, an integer $q$-flow is a map $\phi:E\to\mathbb Z$ with $|\phi(e)|<q$ and \[ \sum_{h\text{ incident with }v} \sigma(h)\phi(e(h))=0 \qquad\text{for every vertex }v, \] where $e(h)$ is the edge of the half-edge $h$. The flow is nowhere-zero if $\phi(e)\neq0$ for every edge. A bidirected graph is flow-admissible if it has a nowhere-zero integer $q$-flow for some integer $q$.

Bouchet's conjecture states that every flow-admissible finite bidirected graph has a nowhere-zero integer $6$-flow, with values in $\{-5,-4,-3,-2,-1,1,2,3,4,5\}$. The bidirected incidence signs are fixed as part of the input and must be respected by the smaller flow.

The premise is existence of an integer flow of some size, not just absence of bridges in the underlying undirected graph. Unlike ordinary orientations, an edge can contribute with the same sign at both endpoints; the half-edge definition includes this distinction explicitly.

Short English statement

Whenever a bidirected graph admits any nonzero integer flow, can it admit one with every edge magnitude at most five?

Research attempt

Scope and method.

This research attempt by GPT-6 Astra Ultra is dated 1 October 2026. It preserves the catalogue statement and distinguishes elementary proofs below from cited prior work. No novelty claim or formal proof-assistant verification is made. The final paragraph states the precise remaining scope.

Literature and the incidence convention.

DeVos [R1] proves a universal twelve-flow bound for the same bidirected integer-flow framework. The present attempt checks elementary cases and obstructions at the sharper six-flow target. Write $B$ for the incidence matrix whose column contains the two half-edge signs; for a loop they are added in the same row. Conservation is exactly $B\phi=0$ over $\mathbb Z$. A loop with equal half-edge signs contributes $\pm2\phi(e)$, so treating every loop as a zero column would change the problem.

Switching and balanced inputs.

Multiplying one row of $B$ by $-1$ reverses all half-edges at that vertex and preserves its integer kernel. Multiplying an edge column by $-1$ corresponds to replacing the flow value on that edge by its negative, preserving magnitudes. Give a non-loop edge sign $s_e=-\sigma(h_1)\sigma(h_2)$, so positive edges have opposite incidences. Vertex switching changes the sign of every non-loop edge incident there.

Suppose the product of edge signs is positive around every cycle, with loops interpreted as one-edge cycles. Choose a spanning tree in each component and switch vertices recursively to make its edges positive. Every remaining edge then is positive as well: its fundamental cycle has positive product and all the other edges on it are positive. Positive loops already have opposite incidences. Thus a balanced input can be switched to an ordinary directed graph. If its underlying graph has only even degrees, orient Euler tours and assign unit values. Edge-column reversals convert this convenient orientation to the switched input orientation; inverse row switching returns to the original input. This proves a nowhere-zero two-flow for balanced Eulerian inputs, while respecting the original fixed half-edge signs through the transformations.

A bridge can carry a nonzero signed flow.

Take vertices $u,v$ connected by a path of any positive length. Put a loop with two incoming half-edges at $u$ and a loop with two outgoing half-edges at $v$. Direct the path from $u$ to $v$. Assign value one to each loop and value two to every path edge. At $u$ the equation is $2\cdot1-2=0$; at $v$ it is $2-2\cdot1=0$; all internal path vertices have one incoming and one outgoing value two. This is a nowhere-zero three-flow even though every path edge is an underlying bridge. The example illustrates precisely why ordinary bridge arguments cannot be imported into the signed conjecture.

An unbalanced cycle obstruction.

A single cycle with negative sign product has no nowhere-zero integer flow. At each degree-two vertex conservation makes the two incident values equal up to sign. Propagating these equations around the cycle returns $\phi(e)=-\phi(e)$, so $\phi(e)=0$ over the integers and then every edge value is zero. For a single negative loop this is just $2\phi(e)=0$. The corresponding modular equation can have nonzero solutions in a group with two-torsion, demonstrating that group flows cannot be substituted for the requested integer flow without a lifting argument. These obstructed graphs fail the flow-admissibility premise; they are not counterexamples to Bouchet's conjecture.

Where summing signed circuits fails.

The path-and-loops example contains unavoidable magnitude two on its connecting path. Summing many such elementary circulations can cancel on common edges or accumulate values exceeding five. Choosing rapidly increasing weights prevents some cancellations but gives no fixed upper bound. A proof for arbitrary admissible inputs must coordinate overlaps and signs; the balanced Euler-tour proof removes exactly the obstruction that makes the general case difficult.

Verification and final status.

The root batch script builds the signed incidence matrices for path lengths one through twenty and verifies the explicit one/two certificates over the integers. It checks both loop incidences separately. UNRESOLVED. Balanced Eulerian inputs and the signed handcuff family above satisfy the target. We have not reduced an arbitrary flow-admissible signed graph to these cases while keeping all magnitudes at most five.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3218 (OPG-131), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Bouchet's 6-flow conjecture. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/bouchets_6_flow_conjecture.
• R1 Matt DeVos, Flows on Bidirected Graphs, arXiv:1310.8406. Primary abstract checked 1 October 2026. https://arxiv.org/abs/1310.8406.