On the existence of disjoint cycles in a graph (Q1307442)

For a graph \(G\) let \(\sigma_2\) be the minimum of the sums \(d_G(x)+d_G(y)\), where \(x\) and \(y\) are nonadjacent vertices of \(G\), and \(d_G(x)\) stands for the degree of \(x\) in \(G\). The main result of the paper is a new proof of the following theorem of \textit{P. Justesen} [Ann. Discrete Math. 41, 299-305 (1989; Zbl 0673.05065)]: Suppose \(| G| =n \geq 3k\) and \(\sigma_2(G) \geq 4k-1\). Then \(G\) contains \(k\) vertex-disjoint cycles.