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

Abstract

If $n$ is even, the $n$-dimensional hypercube can be decomposed into edge-disjoint cycles of length $2^i$ for every value of $i$ from $2$ to $n$.

Figures (5)

• Figure 1: Left: A representation of $Q_4$ as the Cartesian product $C_4 ~\square~ C_4$. With the vertices of $C_4$ labeled $00, 01, 11, 10$, the vertex labels for $Q_4$ are obtained by concatenating the column label of the vertex with the row label of the vertex. For example the large white vertex has label 1110. Center: A partitionable decomposition of $Q_4$ into cycles of length 8. One partition set ${\cal F}_1$ contains the two solid red cycles, while the other partition set ${\cal F}_2$ contains the two dashed blue cycles. Right: A partitionable decomposition of $Q_4$ into cycles of length 16. One partition set ${\cal F}_1$ contains the solid red cycle, while the other partition set ${\cal F}_2$ contains the dashed blue cycle.
• Figure 2: A representation of $Q_6 = Q_4 ~\square~ C_4$. The $Q_4$ whose vertex labels end with 00, 01, 11, and 10 are pictured in the upper left, upper right, lower right, and lower left, respectively. The label for each vertex in $Q_6$ is obtained by concatenating the column label and then the row label of the vertex, followed by these last two coordinates. For example, the address of the large white vertex in the lower left is 001110. All horizontal edges (edges that differ in the first four coordinates) are shown, but the eight curved edges are the only vertical edges shown, and the rest are omitted for clarity. The dashed red and dotted blue cycles are $C^1$ and $C^2$ in the decomposition of $Q_6$ described in the proof of Proposition \ref{['q6']}. The remaining gray, brown, yellow, and omitted vertical edges are partitioned between the graphs $G_B$ and $G_Y$ in Figure \ref{['Q_6 C_32 Decomp']} (top and middle). The brown edges are $\{010100, 110100\}$, $\{010110, 110110\}$, $\{010111, 110111\}$, and $\{010101, 110101\}$. The yellow edges are $\{011000, 111000\}$, $\{011010, 111010\}$, $\{011011, 111011\}$, and $\{011001, 111001\}$.
• Figure 3: Top: The subgraph $G_B$ of $Q_6$. The curved brown edges are the same as the brown edges in Figure \ref{['Q_6C_32']}. The black edges are either gray edges or omitted vertical edges in Figure \ref{['Q_6C_32']}. The four vertical "missing" edges in this figure are the dashed blue curved edges in Figure \ref{['Q_6C_32']}. Middle: The subgraph $G_Y$ of $Q_6$. The curved yellow edges are the same as the yellow edges in Figure \ref{['Q_6C_32']}. The black edges are either gray edges or omitted vertical edges in Figure \ref{['Q_6C_32']}. The four vertical "missing" edges in this figure are the dashed curved red edges in Figure \ref{['Q_6C_32']}. Bottom: A Hamiltonian decomposition for $G_B$ or $G_Y$.
• Figure 4: Top left: The partitionable decomposition of $C_{8} ~\square~ C_4$ into cycles of length four from the proof of Lemma \ref{['8ell']} ($\ell=2$). Top right: Potential recoloring locations $S_1, \ldots, S_8$. Bottom left: A partitionable decomposition of $C_{8} ~\square~ C_4$ into cycles of length 8. Since $n = 2$ the recolored cycles are $L = \{S_1, S_3, S_5, S_7\}$. Bottom right: A partitionable decomposition of $C_{8} ~\square~ C_4$ into cycles of length 32. Since $n = 8$, the recolored cycles are $L = \{S_1, S_2, S_3, S_4, S_5, S_6, S_7\}$.
• Figure 5: Top: The partitionable decomposition of $C_{24} ~\square~ C_4$ into cycles of length four from the proof of Lemma \ref{['8ell']} ($\ell = 6$). Top middle: Potential recoloring locations $S_1, \ldots, S_{24}$. Bottom middle: A partitionable decomposition of $C_{24} ~\square~ C_4$ into cycles of length 24. Since $n=6$, $S_i$ is recolored unless $i \in \{6, 12, 18, 24\}$. Bottom: A partitionable decomposition of $C_{24} ~\square~ C_4$ into cycles of length 96. Since $n=24$, $S_i$ is recolored unless $i =24$.

Theorems & Definitions (14)

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• Theorem \oldthetheorem: Kotzig, 1973
• Theorem \oldthetheorem: Aubert and Schneider, 1982
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