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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.

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