On vertex-disjoint cycles and degree sum conditions

This paper considers a degree sum condition sufficient to imply the existence of k vertexdisjoint 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. © 2017 Elsevier B.V. All rights reserved.