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\thispagestyle{empty} \begin{document} \noindent % % {\bf Uwe
Schauz} % % \medskip \noindent % % {\bf Colorings and Nowhere-
Zero Flows of Graphs in Terms of Berlekamp's Switching Game} % %
\vskip 5mm \noindent % % % % We work with a unifying linear
algebra formulation for nowhere-zero flows and colorings of
graphs and matrices. Given a subspace (code)
$U\leq{\mathbb{Z}_k^n}$ -- e.g.\ the bond or the cycle space
over ${\mathbb{Z}}_k$ of an oriented graph -- \,we call a
nowhere-zero tuple $f\in{\mathbb{Z}_k^n}$ a \emph{flow} of $U$
if $f$ is orthogonal to $U$\!. In order to detect flows, we view
the subspace $U$ as a light pattern on the \(n\)-dimensional
\emph{Berlekamp Board} ${\mathbb{Z}_k^n}$ with $k^n$ light
bulbs. The lights corresponding to elements of $U$ are
\emph{ON}, the others are \emph{OFF}. Then we allow axis-
parallel switches of complete rows, columns, etc. The core
result of