Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An edge-coloured version of Dirac's theorem   Let $G$ be an edge-coloured graph. The minimum colour degree $ \delta^c(G) $
of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at
least $k$ distinct colours on edges incident to $v$. We say that $G$ is
properly coloured if no two adjacent edges have the same colour. In this paper,
we show that every edge-coloured graph $G$ with $ \delta^c(G) \ge 2|G| / 3 $
contains a properly coloured $2$-factor. Furthermore, we show that for any $
\varepsilon > 0 $ there exists an integer $ n_0 $ such that every edge-coloured
graph $G$ with $|G| = n \ge n_0 $ and $ \delta^c(G) \ge ( 2/3 + \varepsilon ) n
$ contains a properly coloured cycle of length $\ell$ for every $3 \le \ell \le
n$. This result is best possible in the sense that the statement is false for $
\delta^c(G) < 2n / 3 $.

32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin