Pósa,s Conjecture
المؤلف:
Bollobás, B.
المصدر:
Extremal Graph Theory. New York: Academic Press, 1978.
الجزء والصفحة:
...
2-3-2022
3047
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.
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