Zaslavsky's edge-disjoint negative-cycle conjecture

Let Σ\Sigma be a signed graph. Denote by p−(Σ)p^-(\Sigma) the maximum number of edge-disjoint negative cycles, by l(Σ)l(\Sigma) its frustration index, and by p(Σ)p(\Sigma) the maximum number of edge-disjoint cycles in its underlying graph. Zaslavsky's edge-disjoint negative-cycle conjecture. One always has

p−(Σ)=min⁡(l(Σ),p(Σ)).p^-(\Sigma)=\min\bigl(l(\Sigma),p(\Sigma)\bigr).

This asserts equality in both general bounds p−(Σ)≤l(Σ)p^-(\Sigma)\leq l(\Sigma) and p−(Σ)≤p(Σ)p^-(\Sigma)\leq p(\Sigma). The supplied text does not give evidence that the conjecture has been resolved, so its status is left open.

References

Primary source

Maximilien Gadouleau and Huiying Zeng, “On the maximum and negative frustration indices of graphs”, arXiv:2606.11108 (2026).

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