Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extensions of Erd\H{o}s-Gallai Theorem and Luo's Theorem with
  Applications   The famous Erd\H{o}s-Gallai Theorem on the Tur\'an number of paths states
that every graph with $n$ vertices and $m$ edges contains a path with at least
$\frac{2m}{n}$ edges. In this note, we first establish a simple but novel
extension of the Erd\H{o}s-Gallai Theorem by proving that every graph $G$
contains a path with at least $\frac{(s+1)N_{s+1}(G)}{N_{s}(G)}+s-1$ edges,
where $N_j(G)$ denotes the number of $j$-cliques in $G$ for $1\leq
j\leq\omega(G)$. We also construct a family of graphs which shows our extension
improves the estimate given by Erd\H{o}s-Gallai Theorem. Among applications, we
show, for example, that the main results of \cite{L17}, which are on the
maximum possible number of $s$-cliques in an $n$-vertex graph without a path
with $l$ vertices (and without cycles of length at least $c$), can be easily
deduced from this extension. Indeed, to prove these results, Luo \cite{L17}
generalized a classical theorem of Kopylov and established a tight upper bound
on the number of $s$-cliques in an $n$-vertex 2-connected graph with
circumference less than $c$. We prove a similar result for an $n$-vertex
2-connected graph with circumference less than $c$ and large minimum degree. We
conclude this paper with an application of our results to a problem from
spectral extremal graph theory on consecutive lengths of cycles in graphs.

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The categories of this paper are: comp-gas
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci