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{\bf Uwe Schauz}
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{\bf Colorings and Nowhere-Zero Flows of Graphs in Terms of Berlekamp's Switching Game}
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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 this paper is that the subspace $U$ has a flow if and only
if the light pattern $U$ cannot be switched off. In particular, a
graph $G$ has a nowhere-zero \(k\)-flow if and only if the
\({\mathbb{Z}}_k\)-bond space of $G$ cannot be switched off. It has
a vertex coloring with $k$ colors if and only if a certain
corresponding code over ${\mathbb{Z}}_k$ cannot be switched
off. Similar statements hold for Tait colorings, and for nowhere-zero
points of matrices. Studying different normal forms to equivalence
classes of light patterns, we find various new equivalents, e.g., for
the Four Color Problem, Tutte's Flow Conjectures and Jaeger's
Conjecture. Two of our equivalents for colorability and existence of
nowhere zero flows of graphs include as special cases results by
Matiyasevich, by Bal\'azs Szegedy, and by Onn. Alon and Tarsi's
sufficient condition for \(k\)-colorability also arrives, remarkably,
as a generalized full equivalent.
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