Degree sum conditions for vertex-disjoint cycles passing through specified vertices

Let k be a positive integer, and let G be a graph of order n≥3k and S be a set of k vertices of G. In this paper, we prove that if σ2(G)≥n+k−1+Δ(G[S]), then G can be partitioned into k vertex-disjoint cycles C1,…,Ck−1,Ck such that |V(Ci)∩S|=1 for 1≤i≤k, and |V(Ci)|=3 for 1≤i≤k−1−Δ(G[S]) and |V(Ci)|≤4 for k−Δ(G[S])≤i≤k−1, where σ2(G) denotes the minimum degree sum of two non-adjacent vertices in G and Δ(G[S]) denotes the maximum degree of the subgraph of G induced by S. This is a common generalization of the results obtained by Dong (2010) and Chiba et al. (2010), respectively. In order to show the main theorem, we further give other related results concerning the degree conditions for the existence of k vertex-disjoint cycles in which each cycle contains a vertex in a specified vertex subset.