Disjoint Small Cycles in Graphs and Bipartite Graphs

Abstract

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.

Supported by the National Natural Science Foundation of China (Grant No. 11161035), Ningxia Ziran (Grant No. NZ1153) and research grant from Ningxia University (Grant No. ndzr10-19).