Vertex-Disjoint Cycles in Graphs

Recently, Korándi et al. (2018) found that if the complement of a graph G is K_k^m-free, then one can always find k - 1 vertex-disjoint cycles covering all but at most 2k^2 m + k^3 vertices. The objective of our research is to find the smallest integer l(k,m) such that k - 1 cycles cover all but at most l many vertices. We conjecture that in general l(k,m)= (k-1)(m-1) and we prove the conjecture when m = 2. We accomplished this by applying mathematical induction and a Posa path rotation-extension technique to show that all but at most k - 1 many vertices are covered by the k - 1 cycles when the order of G is sufficiently large, else we can cover all but at most k many vertices.