  In this paper, we derive generalized Bern-Carrasco-Johansson relations for
color-ordered Yang-Mills amplitudes by imposing gauge invariance conditions and
dimensional reduction appropriately on the new discovered graphic expansion of
Einstein-Yang-Mills amplitudes. These relations are also satisfied by
color-ordered amplitudes in other theories such as color-scalar theory,
bi-scalar theory and nonlinear sigma model (NLSM). As an application of the
gauge invariance induced relations, we further prove that the three types of
BCJ numerators in NLSM , which are derived from Feynman rules, Abelian Z-theory
and Cachazo-He- Yuan formula respectively, produce the same total amplitudes.
In other words, the three distinct approaches to NLSM amplitudes are equivalent
to each other.
