Regular Graphs Without Isomorphic Bisections

A graph is bisectable if its vertex set can be split into two sets inducing isomorphic subgraphs. Ando conjectured that every cubic graph is bisectable, and Das, Pokrovskiy, and Sudakov proved it for large enough connected cubic graphs. We give a constructive proof that, for every $d\geq5$, there exists a non-bisectable $d$-regular graph of even order.