Almost Vertex Bipancyclic Graphs

Abstract

A connected balanced bipartite graph $G$ on $2n$ vertices is almost vertex bipancyclic (i.e., $G$ has cycles of length $6,8,\dots ,2n$ 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 $d(x)+d(y)\ge n+1$. Furthermore, all graphs except $C_4$ on $2n$ ($n\ge 3$) vertices satisfying $P(n)$ are bipancyclic (i.e., there are cycles of length $4,6,\dots ,2n$ in the graph).