Uniquely Bipancyclic Graphs on More than 30 Vertices

Abstract

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 $2\le m\le \frac{n}{2}$. In this note, using computer programs, we show that if $32\le n\le 56$, and $n\ne 44$, 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.