Embedding perfectly balanced 2-caterpillar into its optimal hypercube
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A long-standing conjecture on spanning trees of a hypercube states that a balanced tree on 2^n vertices with maximum degree at most 3 spans the hypercube of dimension n <cit.>. In this paper, we settle the conjecture for a special family of binary trees. A 0-caterpillar is a path. For k≥ 1, a k-caterpillar is a binary tree consisting of a path with j-caterpillars (0≤ j≤ k-1) emanating from some of the vertices on the path. A k-caterpillar that contains a perfect matching is said to be perfectly balanced. In this paper, we show that a perfectly balanced 2-caterpillar on 2^n vertices spans the hypercube of dimension n.
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