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Embedding a subclass of trees into hypercubes. (English) Zbl 1223.05020
Summary: A long standing conjecture of I. Havel [Čas. Pěst. Mat. 109, 135–152 (1984; Zbl 0544.05057)] states that every equipartite tree with maximum degree 3 on \(2^n\) vertices is a spanning subgraph of the \(n\)-dimensional hypercube. The conjecture is known to be true for many subclasses of trees. I. Havel and P. Liebl [J. Graph Theory 10, No. 1, 69-77 (1986; Zbl 0589.05031)] showed that every equipartite caterpillar with maximum degree 3 and having \(2^n\) vertices is a spanning subgraph of the \(n\)-dimensional hypercube. Subsequently, I. Havel [Topics in combinatorics and graph theory. Essays in honour of Gerhard Ringel, 353–358 (1990; Zbl 0743.05016)] remarked that the problem of verification of the conjecture for subdivisions of caterpillars with maximum degree 3 has remained open. In this paper, we show that a subdivision of a caterpillar with \(2^n\) vertices and maximum degree 3 is a spanning subgraph of the \(n\)-dimensional hypercube if it is equipartite and has at most \(n - 3\) vertices on the spine. The problem of embedding such trees that have spines of arbitrary length is still open.

05C05 Trees
05C65 Hypergraphs
05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
Full Text: DOI
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