Weak cycle partition involving degree sum conditions

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 - degenerated cycle partition (abbreviated to k -DCP) 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 . If, in addition, for all i , 1 ≤ i ≤ k , H i is a cycle or K 1 , 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 ≅ C 5 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 3 n + 6 k − 5 4 , then either G has a k -WCP or k = 2 and G is a subgraph of K 2 ∪ K n − 2 ∪ { e } , where e is an edge connecting V ( K 2 ) and V ( K n − 2 ) . By using this, we improve Enomoto and Li’s result for n ≥ max { k + 12 , 10 k − 9 } .