Matchings of quadratic size extend to long cycles in hypercubes

Abstract

Ruskey and Savage in 1993 asked whether every matching in a hypercube can be extended to a Hamiltonian cycle. A positive answer is known for perfect matchings, but the general case has been resolved only for matchings of linear size.

In this paper we show that there is a quadratic function q(n) such that every matching in the n-dimensional hypercube of size at most q(n) may be extended to a cycle which covers at least 3/4 of the vertices.

Keywords

• Gray code
• Hamiltonian cycle
• hypercube
• long cycle
• matching
• Ruskey and Savage problem