Weak cycle partition involving degree sum conditions

AbstractLet G be a graph of order n and k a positive integer. A set of subgraphs H={H1,H2,…,Hk} is called a k-degenerated cycle partition (abbreviated to k-DCP) of G if H1,…,Hk are vertex disjoint subgraphs of G such that V(G)=⋃i=1kV(Hi) and for all i, 1≤i≤k, Hi is a cycle or K1 or K2. If, in addition, for all i, 1≤i≤k, Hi is a cycle or K1, then H is called a k-weak cycle partition (abbreviated to k-WCP) of G. 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-DCP, except G≅C5 and k=2. We prove that if G is a graph of order n≥k+12 that has a k-DCP and if the degree sum of any pair of nonadjacent vertices is at least 3n+6k−54, then either G has a k-WCP or k=2 and G is a subgraph of K2∪Kn−2∪{e}, where e is an edge connecting V(K2) and V(Kn−2). By using this, we improve Enomoto and Li’s result for n≥max{k+12,10k−9}