On vertex-disjoint cycles and degree sum conditions

This paper considers a degree sum condition sufficient to imply the existence of k vertex-disjoint cycles in a graph G. For an integer t≥1, let σt(G) be the smallest sum of degrees of t independent vertices of G. We prove that if G has order at least 7k+1 and σ4(G)≥8k−3, with k≥2, then G contains k vertex-disjoint cycles. We also show that the degree sum condition on σ4(G) is sharp and conjecture a degree sum condition on σt(G) sufficient to imply G contains k vertex-disjoint cycles for k≥2.