Let M be a 3-connected non-graphic binary matroid. Define Y(M) to be the set of elements of M avoiding more than r^(M)-1 non-separating cocircuits, and let widetildeY(M):=E(M)-Y(M). Here r^_M denotes the rank function of the dual matroid. Cocircuit-rank conjecture. One has r^_M(widetildeY(M))≤ 2. This conjecture generalizes the paper's principal theorem, which proves stronger conclusions for non-regular matroids and for regular matroids under specified minor hypotheses. The paper reports a theoretical reduction but says that the computational verification remains to be completed.

Cocircuit-rank conjecture for non-graphic binary matroids

Submitted on 25 November 2012, the original paper asks whether every $3$ -connected non-graphic binary matroid satisfies $r^*_M(\widetilde{Y}(M))\le 2$ . Its theoretical reduction is conditional on a restricted computational verification, which the paper described as still being prepared.

Known results

For a $3$ -connected non-regular binary matroid, $|\widetilde{Y}(M)|\le 1$ .
For regular matroids under the stated $M^*(K_{3,3}^{\prime\prime\prime})$ -minor or $M^*(K_5)$ -minor hypotheses, $\widetilde{Y}(M)=\emptyset$ .
In those regular cases, respectively, $E(M)=Y(M)$ .
A binary $3$ -connected matroid with a $PG(3,2)^*$ -minor satisfies $Y(M)=E(M)$ ; no later proof or counterexample was found.

Current status (as of September 2026): the conjecture remains open; only the original theoretical reduction and partial results are publicly recorded, with the required computational verification unreported.

Cocircuit-rank conjecture for non-graphic binary matroids

Primary source

Progress summary

Solutions 0