Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hamiltonian cycles in 7-tough $(P_3\cup 2P_1)$-free graphs   The toughness of a noncomplete graph $G$ is the maximum real number $t$ such
that the ratio of $|S|$ to the number of components of $G-S$ is at least $t$
for every cutset $S$ of $G$, and the toughness of a complete graph is defined
to be $\infty$. Determining the toughness for a given graph is NP-hard.
Chv\'{a}tal's toughness conjecture, stating that there exists a constant $t_0$
such that every graph with toughness at least $t_0$ is hamiltonian, is still
open for general graphs. A graph is called $(P_3\cup 2P_1)$-free if it does not
contain any induced subgraph isomorphic to $P_3\cup 2P_1$, the disjoint union
of $P_3$ and two isolated vertices. In this paper, we confirm Chv\'{a}tal's
toughness conjecture for $(P_3\cup 2P_1)$-free graphs by showing that every
7-tough $(P_3\cup 2P_1)$-free graph on at least three vertices is hamiltonian.

16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th