Color-kinematics relation from the Feynman diagram perspective

Feynman diagrams for gluon tree amplitudes are studied in the gauge and any number of spacetime dimensions. The color-kinematics combinations $\Delta=n_s-n_t-n_u$ numerators explicitly calculated $N=4,5,6$ gluons to see whether relation $\Delta=0$ is satisfied. This a tedious task because presence four-gluon vertices, large diagrams, numerators, $\Delta$ involved, especially when $N=6$. For on-shell amplitudes, it found that $N=4$, but $\Delta\not=0$ $N=5$ $N=6$ owing vertex. However, {\it local} generalized transformation can bring about $N=5$, not raises question satisfying contain non-local interactions.