A. Three-point tree amplitude B. Four-point tree amplitudes The only nontrivial three-point partial amplitude needed is A (1$_{g}$ , 2$_{g}$ , 3$_{h}$). To give this amplitude, we need to calculate the amplitudes A (1$_{g}$ , 2$_{g}$ , 3$_{g}$ , 4$_{g}$) in which 3$_{g}$ and 4$_{g}$ corre- spond to the left- and right-moving sectors of the graviton 3$_{h}$. Using the color-ordered Feynman rules in [24] and replacing the momenta k$_{3}$ and k$_{4}$ by 1 $_{2}$k$_{3}$ , we can get the ampli- tude A (1$_{g}$ , 2$_{g}$ , 3$_{g}$ , 4$_{g}$). After substituting it into the relation (6a), the three-point amplitude A (1$_{g}$ , 2$_{g}$ , 3$_{h}$) is given where the physical conditions ϵ$_{1}$ · k$_{1}$ = ϵ$_{2}$ · k$_{2}$ = 0, ϵ$_{3}$$_{ρσ}$k ρ 3 = ϵ$_{3}$$_{ρσ}$k σ 3 = 0, momentum conserva- tion k µ $_{1}$+ k µ $_{2}$+ k µ 3 = 0 and the traceless condition of the polarization tensor of graviton ϵ ρ 3 ρ = 0 have been used. This is same with that given by KLT relation(2a). Thus, D$_{2}$ relation can give the three-point tree amplitude where gluons are minimally coupled to graviton. In this subsection, we study the four-point amplitudes A (1$_{g}$ , 2$_{g}$ , 3$_{g}$ , 4$_{h}$) and A (1$_{g}$ , 2$_{g}$ , 3$_{h}$ , 4$_{h}$). We use the spinor helicity formalism[24–26] to consider the four-point amplitudes. Useful properties of spinor helicity formalism are listed in Appendix A. The two gluons corresponding to a graviton with helicity ± 2 take helicity ± 1. Then the four-point tree amplitudes with all legs of positive helicity can be given by pure-gluon amplitudes with all legs of positive helicity via the relations (6b) and (6c). Because the pure-gluon partial amplitudes in which all gluons take the same helicity vanish, we have 8 A (1$_{g}$ , 2$_{g}$ , 3$_{h}$) ∼ 2[ − ϵ$_{1}$ · ϵ$_{2}$ϵ ρσ 3 k$_{1}$$_{ρ}$k$_{2}$$_{σ}$ + ϵ$_{1}$ · k$_{2}$ϵ$_{3}$$_{ρσ}$k$_{1}$$_{ρ}$ϵ$_{2}$$_{σ}$ + ϵ$_{2}$ · k$_{1}$ϵ ρσ 3 ϵ$_{1}$$_{ρ}$k$_{2}$$_{σ}$ ] , (7) A (1 + g , 2 + g , 3 + g , 4 + $_{h}$) = A (1 + g , 2 + g , 3 + h , 4 + $_{h}$) = 0 . (8)