@MISC{Bezrukov_embeddingternary, author = {Sergej Bezrukov}, title = {Embedding Ternary Trees into the Hypercube}, year = {} }

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Abstract

We consider the complete ternary tree with k levels and its embedding as a subgraph into the binary hypercube of possibly small dimension n. The known from [3] upper bound n 2k + 1 is improved up to n 5k=3 ß 1:66k as k ! 1. 1 Introduction Denote by B n the n-dimensional binary hypercube and by T k;t the complete t-ary tree with k levels. T k;t is a rooted tree, the root of it has degree t and all the other vertices which are not leaves have degree t + 1. We are interested to find the minimal n such that T k;t is a subgraph of B n . This minimal n is called the cubical dimension of T k;t and denoted by dim(T k;t ). The function dim(T k;t ) was studied in a number of papers. Thus in [5],[6] it is proved that kt e dim(T k;t ) (k + 1)t 2 + k \Gamma 1; (1) where e = 2:71::. For small values of k and t the following exact results are known (cf. [4],[5],[1] respectively): dim(T k;2 ) = k + 2; dim(T 2;t ) = ¸ 3t + 1 2 ß ; dim(T 3;t ) ¸ 227t 120 ; as t !1...