Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The maximum number of triangles in graphs without large linear forests   Let $G$ be a graph on $n$ vertices. A linear forest is a graph consisting of
vertex-disjoint paths and isolated vertices. A maximum linear forest of $G$ is
a subgraph of $G$ with maximum number of edges, which is a linear forest. We
denote by $l(G)$ this maximum number. Let $t=\left\lfloor (k-1)/2\right
\rfloor$. Recently, Ning and Wang \cite{boning} proved that if $l(G)=k-1$, then
for any $k<n$ \[ e(G) \leq \max \left\{\binom{k}{2},\binom{t}{2}+t (n - t)+ c
\right\}, \] where $c=0$ if $k$ is odd and $c=1$ otherwise, and the inequality
is tight. In this paper, we prove that if $l(G)=k-1$ and $\delta(G)=\delta$
($\delta<\lfloor k/2 \rfloor$), then for any $k<n$ \[ e(G) \leq \max
\left\{\binom{k-\delta}{2}+\delta(n-k+\delta),\binom{t}{2}+t\left(n-t\right)+c
\right\}. \] When $\delta=0$, it reduces to Ning and Wang's result. Moreover,
let $r_3(G)$ be the number of triangles in $G$. We prove that if $l(G)=k-1$ and
$\delta(G)= \delta$, then for any $k<n$ \[ r_3(G)\leq \max
\left\{\binom{k-\delta}{3}+\binom{\delta}{2}(n-k+\delta),\binom{t}{3}+\binom{t}{2}\left(n-t\right)+d
\right\}. \] where $d=0$ if $k$ is odd and $d=t$ otherwise.

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The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
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The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
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The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph