  In this paper, we provide a thorough study on the expansion of single trace
Einstein-Yang-Mills amplitudes into linear combination of color-ordered
Yang-Mills amplitudes, from various different perspectives. Using the gauge
invariance principle, we propose a recursive construction, where EYM amplitude
with any number of gravitons could be expanded into EYM amplitudes with less
number of gravitons. Through this construction, we can write down the complete
expansion of EYM amplitude in the basis of color-ordered Yang-Mills amplitudes.
As a byproduct, we are able to write down the polynomial form of BCJ numerator,
i.e., numerators satisfying the color-kinematic duality, for Yang-Mills
amplitude. After the discussion of gauge invariance, we move to the BCFW
on-shell recursion relation and discuss how the expansion can be understood
from the on-shell picture. Finally, we show how to interpret the expansion from
the aspect of KLT relation and the way of evaluating the expansion coefficients
efficiently.
