Relations for Einstein–Yang–Mills amplitudes from the CHY representation

We show that a recently discovered relation, which expresses tree-level single trace Einstein-Yang-Mills amplitudes with one graviton and $(n-1)$ gauge bosons as a linear combination of pure Yang-Mills tree amplitudes with n gauge bosons, can be derived from the CHY representation. In addition we show that there is a generalisation, which expresses tree-level single trace Einstein-Yang-Mills amplitudes with r gravitons and $(n-r)$ gauge bosons as a linear combination of pure Yang-Mills tree amplitudes with n gauge bosons. We present a general formula for this case.