  In this work, we investigate the off-shell expansion relation of the
Yang-Mills scalar theory. We have shown explicitly that the single-trace
Berends-Giele currents in the Yang-Mills scalar theory can be decomposed into a
term expressed by a linear combination of bi-adjoint scalar Berends-Giele
currents and one that vanishes under the on-shell limit. We proved that the
bi-adjoint scalar currents, as well as the corresponding coefficients, can be
characterized by a graphic approach that was originally studied in
Einstein-Yang-Mills expansion. Furthermore, we generalized the decomposition to
the multi-trace case through unifying relations and established the connection
both in single-trace and multi-trace graphic descriptions. Finally, we
established the relations between the Yang-Mills currents and the single-trace
Yang-Mills scalar currents choosing special reference orders of the Yang-Mills
graphs.
