Long path and cycle decompositions of even hypercubes

Decomposition of hypercubes into sunlet graphs of order eight

“…5. Axenovich, Offner and Tompkins [3] established that Q n has a C n2 kdecomposition for 2n ≤ n2 k ≤ 2 n n .…”

Decompositions of even hypercubes into cycles whose length is a power of two

“…We now state two propositions on decompositions of Cartesian products. The first proposition is Proposition 8 in [4]. with one partition set F = {G 1 , .…”

“…Beyond Hamiltonian decompositions, Stout [11], Horak, Siran, and Wallis [7], Mollard and Ramras [9], and Wagner and Wild [13] showed that Q n can be decomposed into certain trees. Anick and Ramras [2], and independently Erde [5] proved that if n is odd, then Q n can be decomposed into any path whose length divides the number of edges in Q n and is at most n. Fink [6], Horak, Siran, and Wallis [7], Mollard and Ramras [9], Tapadia, Waphare, and Borse [12], and Axenovich, Offner, and Tompkins [4] proved that when n is even, Q n can be decomposed into cycles of certain lengths, among other results, but a complete characterization of the graphs that can decompose Q n remains an open question, even for paths, trees, and cycles.…”

“…Our goal is to find decompositions of hypercube graphs into cycles whose length is a power of two. The results of Ringel [10], Aubert and Schneider [3], and Axenovich, Offner, and Tompkins [4] show that Q n can be decomposed into relatively long cycles whose length is a power of two, and Tapadia, Waphare, and Borse [12] show that Q n can be decomposed into relatively short cycles whose length is a power of two, but there are no published results for cycles whose length is in between. In this paper we give an elementary proof of the following complete characterization.…”