For any Boolean layer cake P, we may consider its (undirected) cover graph
G with vertex set P and x adjacent to y for 
whenever x
covers y or y covers x in P. When does G admit a Hamiltonian
cycle, i. e., when does there exist a closed path in G visiting each vertex
exactly once? If this is the case, we call P itself Hamiltonian;
concretely, this means that P can be enumerated in sequence 
such that either 
covers 
or is covered by for
all i (taken 
).
While the Hamiltonicity of arbitrary layer cakes apparently has not been
studied, the problem has been considered extensively for 2-layer cakes, and
there
especially for the case of the two maximal layers of a Boolean lattice with an
odd number of atoms. The corresponding cover graphs G are bipartite, and
obviously the two chosen layers must have the same size if there is to be a
Hamiltonian cycle in G. The Hamiltonian conjecture states that
P(i,i+1;2i+1) is Hamiltonian for all 
; it is attributed cyclically
around Dejter, Erdös and Trotter (see [34]) and seems to have been
stated first by Havel in [28]. The generalized version,
considered mainly by Dejter and his students, conjectures that P(i,j;i+j)
is Hamiltonian for any choice of 0<i<j.
The (generalized) conjecture is known to hold for P(i,i+1;2i+1) if i<9, for
P(i,i+2;2i+2) if i<7 and for P(i,i+3;2i+3) in case i=6, see the
introduction of [10]. The technique used by Dejter to obtain a
Hamiltonian cycle in P in these cases is, roughly, to `lift' a Hamiltonian
cycle from a
quotient graph of the covering graph G of P; this quotient graph is induced
by the action of a suitable group 
on G (a so-called `Frucht
diagram'). This approach is developed by Dejter and his students in
[9], [10] and [11].
Turning to P(i,i+1;2i+1) and the original conjecture, the main theme is the
search for so-called matchings in cover graphs of such layer cakes. Recall that
a (perfect) matching in a graph G is a collection M of edges of G
such that each vertex of G is incident with exactly one edge from M (see
also section 10.3 of [48] for matchings in (cover graphs of) ordered
sets).
Any Hamiltonian cycle C in a graph G of even order determines two disjoint
matchings in G in
a natural way: Travelling around G via C, put the edges traversed
alternatively
into two subsets 
and 
; and are evidently disjoint
matchings. This suggests that in order to construct Hamiltonian cycles in the
covering graph of P(i,i+1;2i+1), one might look for matchings in these graphs
and try to piece them together in a suitable way. Matchings in
(the covering graph of) P(i,i+1;2i+1)
may be conveniently visualized as bijections b from the i-sets to the
(i+1)-sets such that 
for any i-set A of
P(i,i+1;2i+1). It is, however, highly nontrivial to describe explicitly
suitable classes of such matchings.
One such class is that of lexicographic matchings. Let 
be
any permutation of the set 
. Order the i-subsets of
as follows: Given two such subsets A and B, write
with 
and
with 
. Let 
. Then put A<B iff 
. Order the
i+1-subsets of in the same way.
A map b from the i-sets to the i+1-sets is now set up in the
following way: 
is 
, and 
is the first i+1-set (in the
order defined above) containing 
and not appearing among
(all subsripts referring to this same order).
As shown in Aigner [3], b turns out to be a bijection for any
permutation . Since obviously 
, and 
for r>1 by construction, we obtain a matching in the cover graph of
P(i,i+1;2i+1), the lexicographic matching induced by .
The main result of [17] is negative: Duffus, Sands and Woodrow
prove that for i>1, the union of two lexicographic matchings will never
form a Hamiltonian cycle in P(i,i+1;2i+1). On the positive side, their
analysis of lexicographic matchings (yielding also a noninductive algorithm
which calculates without reference to )
shows that nonlexicographic matchings exist at all for any i>1.
Such nonlexicographic matchings are provided explicitly by Kierstead and
Trotter in [34] (who call them lexical matchings; every
lexicographic matching is lexical but not vice versa). Let 
the
graph with vertices the i-subsets of and edges between any
pair of disjoint such sets. Kierstead and Trotter conjecture in [34]
that for all i a Hamiltonian cycle in P(i,i+1;2i+1) may be found provided
admits a matching; this latter condition being satisfied when
i>0 is even, e.g., or when 
is even. The crucial fact
here is that any matching in may be lifted to a matching in
P(i,i+1;2i+1), see [13].
A related problem is to find 1-factorizations of P(i,i+1;2i+1), that is, collections of i pairwise disjoints matchings of P(i,i+1;2i+1). It is shown in [34] that 1-factorizations may be found using lexical matchings. Another 1-factorization based on so-called modular matchings is introduced by Duffus, Kierstead and Snevily in [14], and it is shown that these matchings exhibit features strikingly similar to those oflexical matchings. It seems to be open whether modular matchings are of any use in order to produce Hamiltonian cycles.