Abstract
A graph with {±1} labels on the edges is α-balanced if the sum of the labels on each induced cycle is congruent (mod 4) to the related entry of a given vector α. A {0, ±1} matrix is α-balanced if an associated graph with {±1} labels on the edges is α-balanced. The condition of α-balancedness for a matrix turns out to be equivalent to the requirement that certain elementary submatrices have absolute determinant as specified by the entries of α. First the graphs that may be labelled to become α-balanced for a given vector α are characterized. Subsequently a concept of almost representation of matroids is introduced, and necessary and sufficient conditions for a matroid to be almost represented by a matrix are given. Then these results on almost representation are combined with the characterization of α-balancedness to establish a new characterization of GF(3)-representable and (as a special case) regular matroids. A new characterization of totally unimodular matrices is a corollary. These results imply known characterizations of GF(3)-representable and regular matroids by R. Reid and W.T. Tutte, respectively. They also unify W.T. Tutte's work on regular matroids and P. Camion's results for totally unimodular matrices.
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