Transversals and Bipancyclicity in Bipartite Graph Families

A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number vertices in graph. theorem Schmeichel and Mitchem states that for $n \geqslant 4$, balanced on $2n$ which each vertex one color class has degree greater than $\frac{n}{2}$ other at least bipancyclic. We prove a generalization this setting transversals. Namely, we show given family $\mathcal{G}$ graphs common set $X$ with bipartition, $\mathcal G$ minimum class, then there exists cycle $4 \leqslant \ell 2n$ uses most edge G$. also $n$ meeting same conditions, perfect matching exactly