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 $\sigma_t(G) \geq 2kt - t + 1$ for any two integers $k \geq 2$ and $t \geq 5$, then $G$ contains $k$ disjoint cycles.