Discrete Applied Mathematics 116 (2002) 73–102
Optimal embeddings of odd ladders into a hypercube
R. Caha, V. Koubek
Department of Theoretical Informatics, Faculty of Mathematics and Physics, Charles University,
MFF UK, Malostransk
am.25, 118 00 Praha 1, Czech Republic
Received 23 January 1998; revised 11 August 2000; accepted 5 September 2000
An embedding of a graph G into a hypercube of dimension k is called optimal if the number
of vertices of G is greater than 2
. A ladder is a special graph in which two paths of the
same length are connected in such a way that each vertex of the ÿrst one is connected by a
path – called a rung – to its corresponding vertex in the second one. We construct an optimal
embedding for every ladder with rungs of odd sizes greater than 6 into a dense set of a hypercube.
? 2002 Elsevier Science B.V. All rights reserved.
Keywords: A dense set of a hypercube; An embedding into a hypercube; A ladder;
An optimal hypercube
The architecture of parallel computers can be modeled by ÿnite graphs – vertices
of a graph correspond to processors of a computer and its edges correspond to com-
munication lines between processors of a computer. An embedding of a graph into
another (i.e., a one-to-one mapping between sets of vertices that maps edges to edges)
describes the simplest simulation between corresponding computers. Therefore, many
papers concerning parallel computers are devoted to graph–theoretical embedding tech-
niques. From the computer view, embeddings into special classes of graphs modeling
technically realizable parallel computers play an important role. One of such classes of
graphs, widely used for modeling technically realizable parallel computers, is formed
by hypercubes. This has motivated many papers studying classes of graphs embeddable
into hypercubes or, in general, properties of hypercubes.
The class of dichotomic trees was probably the ÿrst class of graphs whose embed-
dings into hypercubes were studied, see . References to numerous other papers on
E-mail addresses: email@example.com (R. Caha), firstname.lastname@example.org (V. Koubek).
0166-218X/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.