Flows in Signed Graphs and Binary Matroids

Definitions and mathematical statement

A signed graph is a graph $G$ equipped with a sign map $\sigma:E(G)\to\{+1,-1\}$. A binary matroid is a matroid representable over ${\mathbb{F}}_2$. For either type of object $X$, write $E(X)$ for its edge set or matroid ground set, respectively, and write $w=(w_e)_{e\in E(X)}$ for the supplied nonnegative weights. The record does not specify the scalar domain of these weights, finiteness assumptions, or graph conventions concerning loops and parallel edges.

Source questions, with unresolved terminology made explicit. Let $\mathcal D$ denote the source-intended class of pairs $(X,w)$, where $X$ is a signed graph or binary matroid and $w$ is a nonnegative weight assignment. Write $C(X,w)$ for the unspecified cut condition, $H(X,w)$ for the unspecified assertion that the weights are Eulerian, and $\mathcal I(X,w)$ and $\mathcal F(X,w)$ for the unspecified sets of feasible integer and fractional flows. These symbols are placeholders for definitions missing from the supplied record; they do not supply a flow model.

In the order given by the supplied statement, the three questions have the following logical form: \[ \begin{aligned} \text{(Q1)}\quad &\forall (X,w)\in\mathcal D,\quad C(X,w)\Longrightarrow\mathcal I(X,w)\ne\varnothing,\\ \text{(Q2)}\quad &\forall (X,w)\in\mathcal D,\quad C(X,w)\Longrightarrow\mathcal F(X,w)\ne\varnothing,\\ \text{(Q3)}\quad &\forall (X,w)\in\mathcal D,\quad \left(C(X,w)\land H(X,w)\right)\Longrightarrow\mathcal I(X,w)\ne\varnothing. \end{aligned} \] The supplied wording asks for an integer flow both in (Q1) and in (Q3), with an additional Eulerian hypothesis only in (Q3). This apparent redundancy is retained. No missing distinction between the questions is inferred.

Formulation limitation. The source record does not define the cut inequalities, flow constraints, capacity or demand interpretation of the weights, permitted flow values, or Eulerian condition, particularly for binary matroids. Consequently, these implications record the supplied quantifier structure but are not a fully specified mathematical conjecture. A precise statement requires the original definitions and hypotheses.

Short English statement

For signed graphs and binary matroids, the supplied record asks whether a cut condition guarantees integer or fractional flows, and whether Eulerian weights guarantee an integer flow under the same cut condition. The necessary definitions are missing, and the two integer-flow questions overlap as written. The supplied status is partially_solved; this draft does not independently verify that assessment.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30001935$, title Flows in Signed Graphs and Binary Matroids, exact source problem code catalogue entry. The supplied record gives proposed year $2011$ and names no individual proposer.
• Supplied report citation: Combinatorial Optimization (2012), Oberwolfach Reports, pp. 3003--3076, DOI 10.4171/owr/2011/53. Source URL and literature-source label "original OWR report": https://doi.org/10.4171/owr/2011/53. The background attributes the problem collection to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source, labelled "flow-cut gaps": https://arxiv.org/abs/1008.2136. No authors, precise bibliographic title, or specific supporting theorem are supplied for this link.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the report records Q1--Q3 for signed graphs and binary matroids, that special signed-graph cases are known, and that the general questions remain unresolved. This is supplied dated curation, not independently verified. The source background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement exact and supplies the verification claim that the official report explicitly asks three flow-existence questions under the cut and Eulerian hypotheses. That verification claim is source metadata; the report has not been independently consulted for this draft.