Degree sum condition for cycle coverings with a given number of pentagons

Let G be a graph of order n = 5 r + s , where r and s are positive integers. For s ≥ 7 , Wang (2007) proved that if δ ( G ) ≥ ( n + r ) / 2 , then G contains r + 1 vertex-disjoint cycles that cover all of its vertices, r of which are pentagons. In this paper, we show that for s ≥ 10 , if σ 2 ( G ) ≥ n + r + 1 , then G also contains r + 1 vertex-disjoint cycles covering all vertices of G , including r pentagons. When s is odd and greater than 9, our result extends Wang’s result under the degree sum condition σ 2 ( G ) ≥ n + r + 1 .