Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 of vertices in the graph. A theorem of
Schmeichel and Mitchem states that for $n \geq 4$, every balanced bipartite
graph on $2n$ vertices in which each vertex in one color class has degree
greater than $\frac{n}{2}$ and each vertex in the other color class has degree
at least $\frac{n}{2}$ is bipancyclic. We prove a generalization of this
theorem in the setting of graph transversals. Namely, we show that given a
family $\mathcal{G}$ of $2n$ bipartite graphs on a common set $X$ of $2n$
vertices with a common balanced bipartition, if each graph of $\mathcal G$ has
minimum degree greater than $\frac{n}{2}$ in one color class and minimum degree
at least $\frac{n}{2}$ in the other color class, then there exists a cycle on
$X$ of each even length $4 \leq \ell \leq 2n$ that uses at most one edge from
each graph of $\mathcal G$. We also show that given a family $\mathcal G$ of
$n$ bipartite graphs on a common set $X$ of $2n$ vertices meeting the same
degree conditions, there exists a perfect matching on $X$ that uses exactly one
edge from each graph of $\mathcal G$.

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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph