The confirmation of a conjecture on disjoint cycles in a graph

In this paper, we prove the following conjecture proposed by Gould, Hirohata and Keller [Discrete Math. submitted]: Let $G$ be a graph of sufficiently large order. If ${\sigma }_t(G)\ge 2kt-t+1$ for any two integers $k\ge 2$ and $t \geq 5$,then$G$contains$k$ disjoint cycles.