Extending Perfect Matchings to Gray Codes with Prescribed Ends
A binary (cyclic) Gray code is a (cyclic) ordering of all binary strings of the same length such that any two consecutive strings differ in a single bit. This corresponds to a Hamiltonian path (cycle) in the hypercube. Fink showed that every perfect matching in the $n$-dimensional hypercube $Q_n$ can be extended to a Hamiltonian cycle, confirming a conjecture of Kreweras. In this paper, we study the "path version" of this problem. Namely, we characterize when a perfect matching in $Q_n$ extends to a Hamiltonian path between two prescribed vertices of opposite parity. Furthermore, we characterize when a perfect matching in $Q_n$ with two faulty vertices extends to a Hamiltonian cycle. In both cases we show that for all dimensions $n\ge 5$ the only forbidden configurations are so-called half-layers, which are certain natural obstacles. These results thus extend Kreweras' conjecture with an additional edge, or with two faulty vertices. The proof for the case $n=5$ is computer-assisted.