Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Minimum k-critical bipartite graphs   We study the problem of Minimum $k$-Critical Bipartite Graph of order $(n,m)$
- M$k$CBG-$(n,m)$: to find a bipartite $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and
$n>m>1$, which is $k$-critical bipartite, and the tuple $(|E|, \Delta_U,
\Delta_V)$, where $\Delta_U$ and $\Delta_V$ denote the maximum degree in $U$
and $V$, respectively, is lexicographically minimum over all such graphs. $G$
is $k$-critical bipartite if deleting at most $k=n-m$ vertices from $U$ creates
$G'$ that has a complete matching, i.e., a matching of size $m$. We show that,
if $m(n-m+1)/n$ is an integer, then a solution of the M$k$CBG-$(n,m)$ problem
can be found among $(a,b)$-regular bipartite graphs of order $(n,m)$, with
$a=m(n-m+1)/n$, and $b=n-m+1$. If $a=m-1$, then all $(a,b)$-regular bipartite
graphs of order $(n,m)$ are $k$-critical bipartite. For $a<m-1$, it is not the
case. We characterize the values of $n$, $m$, $a$, and $b$ that admit an
$(a,b)$-regular bipartite graph of order $(n,m)$, with $b=n-m+1$, and give a
simple construction that creates such a $k$-critical bipartite graph whenever
possible. Our techniques are based on Hall's marriage theorem, elementary
number theory, linear Diophantine equations, properties of integer functions
and congruences, and equations involving them.

36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph