Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Ore-type condition for existence of two disjoint cycles   Let $n_{1}$ and $n_{2}$ be two integers with $n_{1},n_{2}\geq3$ and $G$ a
graph of order $n=n_{1}+n_{2}$. As a generalization of Ore's degree condition
for the existence of Hamilton cycle in $G$, El-Zahar proved that if
$\delta(G)\geq
\left\lceil\frac{n_{1}}{2}\right\rceil+\left\lceil\frac{n_{2}}{2}\right\rceil$
then $G$ contains two disjoint cycles of length $n_{1}$ and $n_{2}$. Recently,
Yan et. al considered the problem by extending the degree condition to degree
sum condition and proved that if $d(u)+d(v)\geq n+4$ for any pair of
non-adjacent vertices $u$ and $v$ of $G$, then $G$ contains two disjoint cycles
of length $n_{1}$ and $n_{2}$. They further asked whether the degree sum
condition can be improved to $d(u)+d(v)\geq n+2$. In this paper, we give a
positive answer to this question. Our result also generalizes El-Zahar's result
when $n_{1}$ and $n_{2}$ are both odd.

17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom