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Abstract.Some graph-theoretical tools are used to introduce the concept of "regular genus" §(M"), for every closed n-dimensional PL-manifold M". Then it is proved that the regular genus of every surface equals its genus, and that the regular genus of every 3-manifold Af j equals its Heegaard genus, if M3 is orientable, and twice its Heegaard genus, if M3 is nonorientable. A geometric approach, and some applications in dimension four are exhibited.