Two-versus-one cycle conjecture in sublinear MPC

Prove or refute the conjecture that distinguishing a single large cycle from a graph consisting of two cycles requires (n) rounds in the sublinear Massively Parallel Computation model.

Background

The paper cites the 2-vs-1 cycle problem as a fundamental conditional barrier for sublinear MPC algorithms. The input graph is distributed across machines, and the task is to distinguish whether it is one large cycle or two disjoint cycles. The conjectured lower bound is used as an example of the difficulty of graph-based inputs compared with geometric inputs, where additional structure can enable faster algorithms.