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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.)
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