Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long Cycles in 1-tough Graphs   In 1952, Dirac proved that every 2-connected graph with minimum degree
$\delta$ either is hamiltonian or contains a cycle of length at least
$2\delta$. In 1986, Bauer and Schmeichel enlarged the bound $2\delta$ to
$2\delta+2$ under additional 1-tough condition - an alternative and more
natural necessary condition for a graph to be hamiltonian. In fact, the bound
$2\delta+2$ is sharp for a graph on $n$ vertices when $n\equiv 1(mod\ 3)$. In
this paper we present the final version of this result which is sharp for each
$n$: every 1-tough graph either is hamiltonian or contains a cycle of length at
least $2\delta+2$ when $n\equiv 1(mod\ 3)$, at least $2\delta+3$ when $n\equiv
2(mod\ 3)$ or $n\equiv 1(mod\ 4)$, and at least $2\delta+4$ otherwise.

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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
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The categories of this paper are: econ