  We study dynamics of Dirac solitons in prototypical networks modeling them by
the nonlinear Dirac equation on metric graphs. Soliton solutions of the
nonlinear Dirac equation on simple metric graphs are obtained. It is shown that
these solutions provide reflectionless vertex transmission of the Dirac
solitons under suitable conditions. The constraints for bond nonlinearity
coefficients, allowing reflectionless transmission over a Y-junction are
derived. The analytical results are confirmed by direct numerical simulations.
