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The q-Analog of the Middle Levels Problem

The well-known middle levels problem is to find a Hammiltonian cycle in the graph induced from the binary Hamming graph $\cH_2(2k+1)$ by the words of weight $k$ or $k+1$. In this paper we define the $q$-analog of the middle levels problem. Let $n=2k+1$ and let $q$ be a power of a prime number. Consider the set of $(k+1)$-dimensional subspaces and the set of $k$-dimensional subspaces of $\F_q^n$. Can these subspaces be ordered in a way that for any two adjacent subspaces $X$ and $Y$, either $X \subset Y$ or $Y \subset X$? A construction method which yields many Hamiltonian cycles for any given $q$ and $k=2$ is presented.