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A note on the cubical dimension of new classes of binary trees. (English) Zbl 1363.05030
Summary: The cubical dimension of a graph $$G$$ is the smallest dimension of a hypercube into which $$G$$ is embeddable as a subgraph. The conjecture of I. Havel [Čas. Pěst. Mat. 109, 135–152 (1984; Zbl 0544.05057)] claims that the cubical dimension of every balanced binary tree with $$2^n$$ vertices, $$n\geq 1$$, is $$n$$. The 2-rooted complete binary tree of depth $$n$$ is obtained from two copies of the complete binary tree of depth $$n$$ by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice a 2-rooted complete binary tree and prove that every such balanced tree satisfies the conjecture of Havel.
##### MSC:
 05C05 Trees 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
##### Keywords:
cubical dimension; embedding; Havel’s conjecture; hypercube
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##### References:
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