Partition of a graph into cycles and vertices

Abstract

Let G be a graph of order n and k a positive integer. A set of subgraphs H = { H 1 , H 2 , … , H k } is called a k- weak cycle partition (abbreviated k-WCP) of G if H 1 , … , H k are vertex disjoint subgraphs of G such that V ( G ) = ⋃ i = 1 k V ( H i ) and for all i, 1 ⩽ i ⩽ k , H i is a cycle or K 1 or K 2 . It has been shown by Enomoto and Li that if | G | = n ⩾ k and if the degree sum of any pair of nonadjacent vertices is at least n - k + 1 , then G has a k-WCP. We prove that if G has a k-WCP and if the minimum degree is at least ( n + 2 k ) / 3 , then G can be partitioned into k subgraphs H i , 1 ⩽ i ⩽ k , where each H i is a cycle or K 1 .