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{\bf Shinya Fujita, Alexander Halperin and Colton Magnant}
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{\bf Long Path Lemma concerning Connectivity and Independence Number}
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We show that, in a $k$-connected graph $G$ of order $n$ with $\alpha(G) = 
\alpha$, between any pair of vertices, there exists a path $P$ joining them 
with $$|P| \geq \min \left\{ n, \frac{(k - 1)(n - k)}{\alpha} + k \right\}.$$  
This implies that, for any edge $e \in E(G)$, there is a cycle containing $e$ 
of length at least $$\min \left\{ n, \frac{(k - 1)(n - k)}{\alpha} + k 
\right\}.$$  Moreover, we generalize our result as follows: for any choice 
$S$ of $s \leq k$ vertices in $G$, there exists a tree $T$ whose set of leaves is 
$S$ with $$|T| \geq \min \left\{ n, \frac{(k - s + 1)(n - k)}{\alpha} + k 
\right\}.$$

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