(Fe81-xCoxGa19)(80)B-20 ribbons, with x = 0, 7, 11, 19, and 23, were made by the single-roller quenching method. The x-ray diffraction patterns showed the A2, D0(19), and L1(2) diffraction peaks coexisting with the amorphous matrix background. When Fe was replaced by Co, i.e., when x was increased, the first crystallization temperature (T-X1) and Curie temperature (T-C1) rose greatly. With a sufficiently heavy load (or weight) stretching the ribbon downward, the magnetization MS was pulled along the vertical direction (i.e., the tension-axis). Then, by applying a horizontal field (H-E) up to 6 KOe, we determined the sign and magnitude of the saturation magnetostriction (lambda(S)). The Young's modulus (E-S) at magnetization saturation and Delta E effect was estimated from the stress-strain curve. We discovered that the stress-strain curve of (Fe81Ga19)(80)B-20 exhibits the auxetic (or abnormal) characteristic with Delta E < 0. For the rest of the (Fe81-xCoxGa19)(80)B-20 with x not equal 0, their stress-strain curves are normal with Delta E > 0. Because the sign of lambda(S) of the A2 phase is opposite to that of the D0(19) and L1(2) phases, lambda(S) of (Fe81-xCoxGa19)(80)B-20 is much reduced. However, Delta E/E-0 is approximately equal to 36% is quite large for (Fe58Co23Ga19)(80)B-20. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4747922]