Partition of a graph into cycles and degenerated cycles

Abstract

Let G be a graph of order n and k any positive integer with k⩽ n. It has been shown by Brandt et al. that if | G|= n⩾4 k and if the degree sum of any pair of nonadjacent vertices is at least n, then G can be partitioned into k cycles. We prove that if the degree sum of any pair of nonadjacent vertices is at least n− k+1, then G can be partitioned into k subgraphs H i , 1⩽ i⩽ k, where H i is a cycle or K 1 or K 2, except G= C 5 and k=2.