Pósa's Conjecture

Dirac (1952) proved that if the minimum vertex degree  for a graph  on  nodes, then  contains a Hamiltonian cycle (Bollobás 1978, Komlós et al. 1996).

In 1962, Pósa conjectured that  contains a square of a Hamiltonian cycle if  (Erdős 1964, p. 159; Komlós et al. 1996), where a graph  contains the square of a Hamiltonian cycle if there is a Hamiltonian cycle  such that , for , 2, ..., .

Komlós et al. (1996) proved that there exists a natural number  such that if a graph  has order  and minimum vertex degree at least , then  contains the square of a Hamiltonian cycle. This proved Pósa's conjecture (Erdős 1964) for sufficiently large . Kierstead and Quintana (1998) proved Pósa's conjecture for graphs  containing a 4-clique .

The conjecture was generalized by Seymour (1974) to state that if , then  contains the th power of a Hamiltonian cycle (Komlós et al. 1996).

REFERENCES

Bollobás, B. Extremal Graph Theory. New York: Academic Press, 1978.

Dirac, G. A. "Some Theorems on Abstract Graphs." Proc. London Math. Soc. 2, 69-81, 1952.

Erdős, P. "Problem 9." In Theory of Graphs and Its Applications, Proceedings of the Symposium held in Smolenice in June 1963 (Ed. M. Fiedler). Prague, Czechoslovakia: Publishing House of the Czechoslovak Academy of Sciences, p. 159, 1964.

Fan, G. and Kierstead, H. A. "Hamiltonian Square-Paths." J. Combin. Theory Ser. B 67, 167-182, 1996.

Kierstead, H. A. and Quintana, J. "Square Hamiltonian Cycles in Graphs with Maximal 4-Cliques." Disc. Math. 178, 81-92, 1998.

Komlós, J.; Sárkőzy, G. N.; and Szemerédi, E. "On the Square of a Hamiltonian Cycle in Dense Graphs." In Random Structures Algorithms 9, 193-211, 1996.

Seymour, P. Problem Section in Combinatorics: Proceedings of the British Combinatorial Conference, 1973 (Ed. T. P. McDonough and V. C. Mavron). Cambridge, England: Cambridge University Press, pp. 201-202, 1974.