Uniquely Bipancyclic Graphs on More than 30 Vertices

A bipartite graph on \(n\) vertices, with \(n\) even, is called uniquely bi-pancyclic (UBPC) if it contains precisely one cycle of length \(2m\) for every \(2 \leq m \leq \frac{n}{2}\). In this note, using computer programs, we show that if \(32 \leq n \leq 56\), and \(n \neq 44\), then there are no UBPC graphs of order \(n\). We also present the six non-isomorphic UBPC graphs of order 44. This improves the recent results on UBPC graphs of order at most 30.