Gray Codes Faulting Matchings

A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2 binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an nbit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Qn, and a cyclic Gray code as a Hamiltonian cycle of Qn. In this paper we study Hamiltonian paths and cycles of Qn avoiding a given set of faulty edges that form a matching, briefly called (cyclic) Gray codes faulting a given matching. Given a matching M and two vertices u, v of Qn, n ≥ 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M , for the existence of a Gray code between u and v faulting M . As a corollary, we obtain a similar characterization for a cyclic Gray code faulting M . In particular, in case that M is a perfect matching, Qn has a (cyclic) Gray code faulting M if and only if Qn − M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Qn can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamiltonicity of Qn with faulty edges, which is NP-complete in general, becomes polynomial for up to 2 edges provided they form a matching.