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Decomposing complete graphs into cubes

Saad I. El-Zanati;  C. Vanden Eynden — 2006

Discussiones Mathematicae Graph Theory

This paper concerns when the complete graph on n vertices can be decomposed into d-dimensional cubes, where d is odd and n is even. (All other cases have been settled.) Necessary conditions are that n be congruent to 1 modulo d and 0 modulo 2 d . These are known to be sufficient for d equal to 3 or 5. For larger values of d, the necessary conditions are asymptotically sufficient by Wilson’s results. We prove that for each odd d there is an infinite arithmetic progression of even integers n for which...

On Decomposing Regular Graphs Into Isomorphic Double-Stars

Saad I. El-Zanati;  Marie Ermete;  James Hasty;  Michael J. Plantholt;  Shailesh Tipnis — 2015

Discussiones Mathematicae Graph Theory

A double-star is a tree with exactly two vertices of degree greater than 1. If T is a double-star where the two vertices of degree greater than one have degrees k1+1 and k2+1, then T is denoted by Sk1,k2 . In this note, we show that every double-star with n edges decomposes every 2n-regular graph. We also show that the double-star Sk,k−1 decomposes every 2k-regular graph that contains a perfect matching.

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