A new result on disjoint cycles in graphs

Let k be a positive integer. For a graph G, we define ‘${\sigma }_1^2\left(G\right)$’ the minimum value of max$\left\{d\right(x),d(y\left)\right\}$ for any pair of nonadjacent vertices x and y. In 1963, Corrádi and Hajnal proved a classical result: every graph G of order at least 3k with minimum degree at least 2k contains k disjoint cycles. Kierstead, Kostochka and Yeager refined Corrádi-Hajnal Theorem by considering the minimum degree of $2k-1$. In this paper, we characterize all graphs on at least $4k+2$ vertices with ${\sigma }_1^2\left(G\right)\ge 2k-1$ without k disjoint cycles.