Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existences of rainbow matchings and rainbow matching covers   Let $G$ be an edge-coloured graph. A rainbow subgraph in $G$ is a subgraph
such that its edges have distinct colours. The minimum colour degree
$\delta^c(G)$ of $G$ is the smallest number of distinct colours on the edges
incident with a vertex of $G$. We show that every edge-coloured graph $G$ on
$n\geq 7k/2+2$ vertices with $\delta^c(G) \geq k$ contains a rainbow matching
of size at least $k$, which improves the previous result for $k \ge 10$.
  Let $\Delta_{\text{mon}}(G)$ be the maximum number of edges of the same
colour incident with a vertex of $G$. We also prove that if $t \ge 11$ and
$\Delta_{\text{mon}}(G) \le t$, then $G$ can be edge-decomposed into at most
$\lfloor tn/2 \rfloor $ rainbow matchings. This result is sharp and improves a
result of LeSaulnier and West.

31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con