Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A polynomial-time algorithm to determine (almost) Hamiltonicity of dense
  regular graphs   We give a polynomial-time algorithm for detecting very long cycles in dense
regular graphs. Specifically, we show that, given $\alpha \in (0,1)$, there
exists a $c=c(\alpha)$ such that the following holds: there is a
polynomial-time algorithm that, given a $D$-regular graph $G$ on $n$ vertices
with $D\geq \alpha n$, determines whether $G$ contains a cycle on at least $n -
c$ vertices. The problem becomes NP-complete if we drop either the density or
the regularity condition. The algorithm combines tools from extremal graph
theory and spectral partitioning as well as some further algorithmic
ingredients.

14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con