The D$_{2}$ relation for a given amplitude may have different expressions as in string theory. A tree amplitude for gauge-gravity minimal coupling can be expressed by different sets of pure-gluon partial tree amplitudes. The permutations of the legs and the factors in different expressions are different. However, the partial tree amplitudes of gluons are not independent of each other, there are relations among the partial tree amplitudes with gluons[6, 30]. Then the different expressions of the D$_{2}$ relation for a given amplitude can be related by the relations among pure-gluon partial tree amplitudes. To see this, we take amplitude with three gluons and one graviton as an example. In Section II, the relation is given by (6b), the factor is in s$_{13}$ -channel. However, in Section IV, the relation is given by (36). For N = 3, we have where we denote the two gluons corresponding to 4$_{h}$ by 4$_{g}$ and 5$_{g}$. Then the amplitude is given by different expressions. In the last expression, s$_{2}$$_{,}$$_{4}$ = s$_{1}$$_{,}$$_{3}$ , s$_{24}$ + s$_{34}$ = − s$_{14}$ = − s$_{23}$ and s$_{34}$ = s$_{12}$ . The second and the third terms can be given by one term with the factor − s$_{23}$ . Using the relations among partial amplitudes, we have Then the two relations (37) and (6b) are equivalent. In this paper, we study the amplitudes where gluons are minimally coupled with gravitons. We find the three- and four-point amplitudes satisfy the field theory limits of D$_{2}$ relations 23 V. CONCLUSION A (1$_{g}$ , 2$_{g}$ , 3$_{g}$ , 4$_{h}$) = g ( − κ 2 ) [ s$_{24}$ A (1$_{g}$ , 5$_{g}$ , 2$_{g}$ , 4$_{g}$ , 3$_{g}$) + s$_{24}$ A (1$_{g}$ , 4$_{g}$ , 2$_{g}$ , 3$_{g}$ , 5$_{g}$) + s$_{34}$ A (1$_{g}$ , 4$_{g}$ , 2$_{g}$ , 3$_{g}$ , 5$_{g}$) + s$_{34}$ A (1$_{g}$ , 2$_{g}$ , 4$_{g}$ , 3$_{g}$ , 5$_{g}$) ] , (37) A (1$_{g}$ , 4$_{g}$ , 2$_{g}$ , 3$_{g}$ , 5$_{g}$) = s$_{13}$ s$_{23}$ A (1$_{g}$ , 5$_{g}$ , 2$_{g}$ , 4$_{g}$ , 3$_{g}$) , A (1$_{g}$ , 2$_{g}$ , 4$_{g}$ , 3$_{g}$ , 5$_{g}$) = s$_{13}$ s$_{12}$ A (1$_{g}$ , 5$_{g}$ , 2$_{g}$ , 4$_{g}$ , 3$_{g}$) . (38)