Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On maximum $k$-edge-colorable subgraphs of bipartite graphs   If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of
colors to edges of $G$ from the set of $k$ colors, so that adjacent edges
receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it
is the largest among all $k$-edge-colorable subgraphs of $G$. For a graph $G$
and $k\geq 0$, let $\nu_{k}(G)$ be the number of edges of a maximum
$k$-edge-colorable subgraph of $G$. In 2010 Mkrtchyan et al. proved that if $G$
is a cubic graph, then $\nu_2(G)\leq \frac{|V|+2\nu_3(G)}{4}$. This result
implies that if the cubic graph $G$ contains a perfect matching, in particular
when it is bridgeless, then $\nu_2(G)\leq \frac{\nu_1(G)+\nu_3(G)}{2}$. One may
wonder whether there are other interesting graph-classes, where a relation
between $\nu_2(G)$ and $\frac{\nu_1(G)+\nu_3(G)}{2}$ can be proved. Related
with this question, in this paper we show that $\nu_{k}(G) \geq
\frac{\nu_{k-i}(G) + \nu_{k+i}(G)}{2}$ for any bipartite graph $G$, $k\geq 0$
and $i=0,1,...,k$.

5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int