Improved degree conditions for 2-factors with cycles in hamiltonian graphs

In this paper, we consider conditions that ensure a hamiltonian graph has a 2-factor with exactly k cycles. Brandt et al. proved that if G is a graph on n ≥ 4 k vertices with minimum degree at least n 2 , then G contains a 2 -factor with exactly k cycles; moreover this is best possible. Faudree et al. asked if there is some c < 1 2 such that δ ( G ) ≥ c n would imply the existence of a 2-factor with k -cycles under the additional hypothesis that G was hamiltonian. This question was answered in the affirmative by Sárközy, who used the regularity–blow-up method to show that there exists some ε > 0 such that for every k ≥ 1 and large n , δ ( G ) ≥ ( 1 2 − ε ) n suffices. rove on this result, giving an elementary proof that for every ε > 0 and k ≥ 1 , if G is a hamiltonian graph on n ≥ 3 k ε vertices with δ ( G ) ≥ ( 2 5 + ε ) n , then G contains a 2-factor with k cycles.