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We investigate color-kinematics duality for gauge-theory amplitudes producedby the pure nonabelian Yang-Mills action deformed by higher-dimensionoperators. For the operator denoted by F^3, the product of three fieldstrengths, the existence of color-kinematic dual representations follows fromstring-theory monodromy relations. We provide explicit dual representations,and show how the double-copy construction of gravity amplitudes based on themis consistent with the Kawai-Lewellen-Tye relations. It leads to the amplitudesproduced by Einstein gravity coupled to a dilaton field phi, and deformed byoperators of the form phi R^2 and R^3. For operators with higher dimensionsthan F^3, such as F^4-type operators appearing at the next order in thelow-energy expansion of bosonic and superstring theory, the situation is morecomplex. The color structure of some of the F^4 operators is incompatible witha simple color-kinematics duality based on structure constants f^{abc}, buteven the color-compatible F^4 operators do not admit the duality. In contrast,the next term in the alpha-prime expansion of the superstring effective action--- a particular linear combination of D^2 F^4 and F^5-type operators --- doesadmit the duality, at least for amplitudes with up to six external gluons.

Color-kinematics duality and double-copy construction for amplitudes from higher-dimension operators