Almost Vertex Bipancyclic Graphs

A connected balanced bipartite graph \(G\) on \(2n\) vertices is almost vertex bipancyclic (i.e., \(G\) has cycles of length \(6, 8, \ldots, 2n\) through each vertex of \(G\)) if it satisfies the following property \(P(n)\): if \(x, y \in V(G)\) and \(d(x, y) = 3\) then \(d(x) + d(y) \geq n + 1\). Furthermore, all graphs except \(C_4\) on \(2n\) (\(n \geq 3\)) vertices satisfying \(P(n)\) are bipancyclic (i.e., there are cycles of length \(4, 6, \ldots, 2n\) in the graph).