kk disjoint cycles containing specified independent vertices

Let k⩾1k⩾1 be an integer and G   be a graph of order n⩾3kn⩾3k satisfying the condition that σ2(G)⩾n+k-1σ2(G)⩾n+k-1. Let v1,…,vkv1,…,vk be k independent vertices of G, and suppose that G has k   vertex-disjoint triangles C1,…,CkC1,…,Ck with vi∈V(Ci)vi∈V(Ci) for all 1⩽i⩽k1⩽i⩽k.Then G has k   vertex-disjoint cycles C1′,…,Ck′ such that(i)vi∈V(Ci′) for all 1⩽i⩽k1⩽i⩽k.(ii)V(C1′)∪⋯∪V(Ck′)=V(G), and(iii)At least k-1k-1 of the k   cycles C1′,C2′,…,Ck′ are triangles.The condition of degree sum σ2(G)⩾n+k-1σ2(G)⩾n+k-1 is sharp.