relations for three- and four-point amplitudes are given as In field theory limits, KLT relations factorize the pure-graviton tree amplitudes into products of tree amplitudes for gluons corresponding to left- and right-moving sectors. The tree amplitudes where gravitons are minimally coupled to gluons can also be factorized by KLT relations. In this situation, the two sectors of a graviton with helicity ± 2 correspond to two gluons with helicity ± 1, while the two sectors of a gluon with helicity ± 1 correspond to one gluon with helicity ± 1 and one scalar particle. The gauge degrees of freedom are taken by the scalar field in one sector. With these correspondences, KLT relations express the amplitudes for N gluons and M gravitons by products of amplitudes in left- and right-moving sectors. The amplitudes in the left-moving sector are pure-gluon partial tree amplitudes with N + M legs while the amplitudes in the right-moving sector are partial tree amplitudes for M gluons and N scalar particles. Using the Feynman rules given in [17], one can calculate the partial amplitudes in the left- and right-moving sectors, then the amplitudes where gluons are minimally coupled with gravitons can be given by KLT relations. The KLT factorization relations in string theory hold on S$_{2}$ . But they do not hold on D$_{2}$ . In D$_{2}$ case, the left- and right-moving sectors of closed strings are connected into a single one[5]. The relations between amplitudes for N open strings in addition to M closed strings on D$_{2}$ and open string tree amplitudes are given as In the relation (3), we do not introduce the Chan-Paton degrees of freedom. On D$_{2}$ , one can add Chan-Paton factor to the ends of open strings to incorporate gauge degree of freedom. 5 M ( N,M ) D$_{2}$ ∼ g N − $^{2}$κ M ∑ P A ( N, 2 M $^{)}$( P ) e iπ Θ $^{′}$( P $^{)}$. (3) M (1 , 2 , 3) ∼ κ A (1 , 2 , 3) ¯ A (1 , 2 , 3) , (2a) M (1 , 2 , 3 , 4) ∼ κ $^{2}$( − i ) s$_{12}$ A (1 , 2 , 3 , 4) ¯ A (1 , 2 , 4 , 3) . (2b)