  We examine the dynamics of an ensemble of phase oscillators that are divided
in $k$ sets, with time-delayed coupling interactions {\em only} between
oscillators in different sets or partitions. The network of interactions thus
form a $k-$partite graph. We observe a variety of phase-locked states that
include, in addition to the in-phase fully synchronized solution, a variety of
splay cluster solutions; all oscillators within a partition are synchronised
and the phase differences between oscillators in different partitions are
multiples of $2\pi/k$. Such solutions exist independent of the delay and we
determine the generalised stability criteria for the existence of these
phase-locked solutions for the $k-$partite system. With increase in time-delay,
there is an increase in multistability: the above generic solutions coexist
with a number of other partially synchronized solutions. We apply the
Ott-Antonsen ansatz for the special case of a symmetric $k-$partite graph to
obtain a single time-delayed differential equation for the attracting
synchronization manifold. The agreement with numerical results for the specific
case of oscillators on a tripartite lattice (the $k=3$ case) is excellent.
