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Optimal embeddings of odd ladders into a hypercube. (English) Zbl 1005.05016
An embedding of a graph $$G$$ into (the graph of) a hypercube of dimension $$k$$ is called optimal if the number of vertices of $$G$$ is greater than $$2^{k-1}$$. A ladder is a graph consisting of two paths of the same length $$n$$ and of $$n+1$$ paths, called rungs, such that the corresponding vertices of the two paths are connected by one of the rungs. Such a ladder is called odd if all its rungs are of odd size.
Continuing their own work (see [Eur. J. Comb. 18, 249-266 (1997; Zbl 0883.05041)]) and that of others (see S. Bezrukov, B. Monien, W. Unger and G. Wechsung [Discrete Appl. Math. 83, 21-29 (1998; Zbl 0906.05019)]) the authors prove that every odd ladder with rungs of sizes greater than 6 has an optimal embedding into a hypercube. An example of an odd ladder with ten rungs of sizes 3 and 5 is given, found by a computer program, which does not have an optimal embedding into a hypercube. It remains open whether each odd ladder with rungs of sizes at least 5 has an optimal embedding into a hypercube. All proofs depend on sophisticated investigations of so-called dense sets in hypercubes.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 68R10 Graph theory (including graph drawing) in computer science
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##### References:
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