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Аннотация:By Steinitz' Theorem all triangulations of a sphere are generated from one triangulation with four vertices by certain sequences of operations called vertex splittings. A theorem of Barnette asserts that all triangulations of the projective plane can be generated from two irreducible triangulations. In the present work we obtain an analogous result for the torus: we show that all triangulations of the torus are generated by 21 irreducible triangulations (they are found explicitly) by applying the same vertex splitting operations. Two tables, one figure.