#### Document Type

Article

#### Publication Date

2011

#### Publication Title

The Electronic Journal of Combinatorics

#### ISSN

1077-8926

#### Abstract

We show that, in a k-connected graph G of order n with α(G)=α, between any pair of vertices, there exists a path P joining them with

|P|≥min{n,(k−1)(n−k)/α +k}.

This implies that, for any edge e∈E(G), there is a cycle containing e of length at least

min{n,(k−1)(n−k)/α +k}.

Moreover, we generalize our result as follows: for any choice S of s≤k vertices in G, there exists a tree T whose set of leaves is S with

|T|≥min{n,(k−s+1)(n−k)/α +k}.

#### Recommended Citation

Fujita, Shinya, Alexander Halperin, Colton Magnant.
2011.
"Long Path Lemma Concerning Connectivity and Independence Number."
*The Electronic Journal of Combinatorics*, 18 (1).
source: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p149

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/116

## Comments

Copyright of the article remains with the author. Article obtained from the

Electronic Journal of Combinatorics.