Disjoint cycles of order at least 5

We prove that if G is a graph of order at least 5k with k ≥ 2 and the minimum degree of G is at least 3k then G contains k disjoint cycles of length at least 5. This supports the conjecture by Wang [Australas. J. Combin. 54 (2012), 59–84]: if G is a graph of order at least (2d+1)k and the minimum degree of G is at least (d+1)k with k ≥ 2 then G contains k disjoint cycles of length at least 2d+ 1.

[1]  P. Erdos,et al.  On the maximal number of independent circuits in a graph , 1963 .

[2]  Hong Wang,et al.  Disjoint 5-cycles in a graph , 2012, Discuss. Math. Graph Theory.

[3]  Hong Wang,et al.  An extension of the Corrádi-Hajnal Theorem , 2012, Australas. J Comb..

[4]  A. Shapira,et al.  Extremal Graph Theory , 2013 .

[5]  Hong Wang.  Proof of the Erdős–Faudree Conjecture on Quadrilaterals , 2010, Graphs Comb..

[6]  Hong Wang,et al.  Vertex-disjoint quadrilaterals in graphs , 2004, Discret. Math..

[7]  Ingo Schiermeyer,et al.  On quadrilaterals in a graph , 1999, Discret. Math..

[8]  Mohamed H. El-Zahar,et al.  On circuits in graphs , 1984, Discret. Math..