Phase-locking in $k$-partite networks of delay-coupled oscillators

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.