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 $\mat...

An independent transversal of a graph G $G$ with vertex partition P ${\mathscr{P}}$ is an set intersecting each block in single vertex. Wanless and Wood proved that if has size at least t $t$ the average degree vertices most ∕ 4 $t\unicode{x02215}4$ , then exists. We present construction showing this result optimal: for any ε > 0 $\varepsilon \gt 0$ sufficiently large there forest into parts su...