Long Path Lemma Concerning Connectivity and Independence Number

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}