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

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 $-K_{4}$.