  In this paper, we first establish well-posedness results for one-dimensional
McKean-Vlasov stochastic differential equations (SDEs) and related particle
systems with a measure-dependent drift coefficient that is discontinuous in the
spatial component, and a diffusion coefficient which is a Lipschitz function of
the state only. We only require a fairly mild condition on the diffusion
coefficient, namely to be non-zero in a point of discontinuity of the drift,
while we need to impose certain structural assumptions on the
measure-dependence of the drift. Second, we study Euler-Maruyama type schemes
for the particle system to approximate the solution of the one-dimensional
McKean-Vlasov SDE. Here, we will prove strong convergence results in terms of
the number of time-steps and number of particles. Due to the discontinuity of
the drift, the convergence analysis is non-standard and the usual strong
convergence order $1/2$ known for the Lipschitz case cannot be recovered for
all schemes.
