Disjoint Small Cycles in Graphs and Bipartite Graphs

Let k be a positive integer and let G be a graph of order n ≥ 3k + 1, X be a set of any k distinct vertices of G. It is proved that if \( d\left( x \right) + d\left( y \right) \ge n + 2k - 2\) for any pair of nonadjacent vertices \(x,y \in V\left( G \right)\), then G contains k disjoint cycles T 1, ⋯ ,T k such that each cycle contains exactly one vertex in X, and \(\left| {{T_i}} \right| = 3\) for each 1 ≤ i ≤ k or \(\left| {{T_k}} \right| = 4\) and the rest are all triangles. We also obtained two results about disjoint 6-cycles in a bipartite graph.