An Optimal Binding Number Condition for Bipancyclism

Abstract

Let $G=(V_1,V_2,E)$ be a balanced bipartite graph with $2n$ vertices. The bipartite binding number of $G$, denoted by $B(G)$, is defined to be $n$ if $G=K_{n,n}$ and $\min_{i\,\in\,\{1,2\}}\,\min_{\emptyset\ne S\subseteq V_i\atop\hfill |N(S)|<n}|N(S)|/|S|$ otherwise. We call $G$ bipancyclic if it contains a cycle of every even length $m$ for $4 \le m \le 2n$. The purpose of this paper is to show that if $B(G)>3/2$ and $n \ge 139$, then $G$ is bipancyclic; the bound $3/2$ is best possible in the sense that there exist infinitely many balanced bipartite graphs $G$ that have $B(G)=3/2$ but are not Hamiltonian.