The case k=2 (conjectured by Pósa in 1962) was obtained in the paper
[J. Komlós, G.N. Sárközy and E. Szemerédi, On the
Square of a Hamiltonian Cycle in Dense Graphs, Random Structures and
Algorithms 9 (1996), 193-211].
Abstract: In 1962 Pósa conjectured that any graph G of
order n and minimum degree at least 2n/3 contains the square of a Hamiltonian cycle. In this paper we prove this conjecture for sufficiently large n.
The general case (conjectured by Seymour in 1974) was obtained by the same
three authors. In his lecture at the INTERNATIONAL COLLOQUIUM ON
EXTREMAL GRAPH THEORY
(Balatonlelle, Hungary, July 27 - August 2, 1997)
Gábor Sárközy presented
some recent results
(due to the three mentioned authors)
on the Regularity Lemma - Blow-up Lemma method.
From the abstract of the talk: A typical example: the Seymour
Conjecture is true for sufficiently large graphs, i.e. for any positive
integer k there is an N(k) such that if a graph G has order n >= N(k)
and the minimum degree is at least kn/(k+1), then G contains the k'th
power of a Hamiltonian cycle.
Bjarne Toft, September 2 1997.