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arXiv · 2606.23826

The excluded minors for $\mathsf{Z}_{3}$-gainable and regular biased graphs

Nick Brettell·Rutger Campbell·Daryl Funk·Dillon Mayhew
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Abstract

We prove that a biased graph is gainable over the group $\mathsf{Z}_{3}$ if and only if it contains no minor isomorphic to $(4K_{2},\emptyset)$, $\pm K_{3}$, or $-K_{4}$. We develop a theory of "partial groups" that is analogous to that of partial fields, and we use this theory to show that a biased graph is gainable over every non-trivial group if and only if it is gainable over $\mathsf{Z}_{2}$ and $\mathsf{Z}_{3}$. From this we derive an independent proof of the theorem due to Gerards that a biased graph is gainable over every non-trivial group if and only if it has no minor isomorphic to $(3K_{2},\emptyset)$, $\pm K_{3}$, or $-K_{4}$.

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Nick Brettell, Rutger Campbell, Daryl Funk, Dillon Mayhew. 2026-06-22. The excluded minors for $\mathsf{Z}_{3}$-gainable and regular biased graphs. https://arxiv.org/abs/2606.23826

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