Bihermitian surfaces with odd first Betti number

Compact bihermitian surfaces are considered, that is, compact, oriented, co nformal four-manifolds admitting two distinct compatible complex structures . It is shown that if the first Betti number is odd then, with respect to e ither complex structure, such a manifold belongs to Class VII of the Enriqu es-Kodaira classification. Moreover, it must be either a special Hopf or an Inoue surface (in the strongly bihermitian case), or obtained by blowing-u p a minimal class VII surface with curves (in the non-strongly bihermitian case).