Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Strengthening theorems of Dirac and Erd\H{o}s on disjoint cycles   Let $k \ge 3$ be an integer, $H_{k}(G)$ be the set of vertices of degree at
least $2k$ in a graph $G$, and $L_{k}(G)$ be the set of vertices of degree at
most $2k-2$ in $G$. In 1963, Dirac and Erd\H{o}s proved that $G$ contains $k$
(vertex-)disjoint cycles whenever $|H_{k}(G)| - |L_{k}(G)| \ge k^{2} + 2k - 4$.
The main result of this paper is that for $k \ge 2$, every graph $G$ with
$|V(G)| \ge 3k$ containing at most $t$ disjoint triangles and with $|H_{k}(G)|
- |L_{k}(G)| \ge 2k + t$ contains $k$ disjoint cycles. This yields that if $k
\ge 2$ and $|H_{k}(G)| - |L_{k}(G)| \ge 3k$, then $G$ contains $k$ disjoint
cycles. This generalizes the Corr\'{a}di-Hajnal Theorem, which states that
every graph $G$ with $H_{k}(G) = V(G)$ and $|H_{k}(G)| \ge 3k$ contains $k$
disjoint cycles.

22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio