Binding number, minimum degree and bipancyclism in bipartite graphs

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 and $\mathop {\min }\limits_{i \in \left\{ {1,2} \right\}} \mathop {\min }\limits_{\O \ne S \subseteq {V_{I,}}\left| {N\left( S \right)} \right| \prec n} \left| {N\left( S \right)} \right|/\left| S \right|$ otherwise. We call G bipancyclic if it contains a cycle of every even length m for 4 m 2n. A theorem showed that if G is a balanced bipartite graph with 2n vertices, B(G) > 3/2 and n 139, then G is bipancyclic. This paper generalizes the conclusion as follows: Let 0 < c < 3/2 and G be a 2-connected balanced bipartite graph with 2n (n is large enough) vertices such that B(G) c and δ(G) (2 - c)n/(3 - c) + 2/3. Then G is bipancyclic.