Minimum-Turn Tours of Even Polyominoes

Let $P$ be a connected bounded region in the plane formed out of $2 \times 2$ blocks joined by their sides. Peng and Rascoussier conjectured that all minimum-turn Hamiltonian cycles of $P$ exhibit a certain regular structure. We prove this conjecture in the special case when $P$ is a topological disk. The proof proceeds in two phases - a "downward" phase where we break apart an irregular Hamiltonian cycle into a collection of shorter cycles; and an "upward" phase where we put it back together in a different way so that, overall, the number of turns in it decreases.