The maximum number of turns of a cycle in a rectangular grid graph

We determine the largest number V(m, n) of turns of a cycle in the grid graph of m × n lattice points for all 2 ≤ m ≤ n. For m ≡ 2 (mod 4), m ≥ 6, it is mn − 2⌈(2n + m − 2)/6⌉ − 2δ, where δ ∈ {0, 1} is 1 exactly when n lies in an explicit set 𝒳m, which is infinite for m = 6 and has at most five elements for m ≥ 10. For odd m it is (m − 1)n, except that V(m, m) = m(m − 1) − 2 for m ≥ 7; in particular, the maximum on the 19 × 19 Go board is 340. For odd m and even n, and for 4 | m, where V(m, n) = m(n − 1), these are the maxima for Hamiltonian cycles found by Tan and Zhang. For Hamiltonian cycles with m ≡ 2 (mod 4), which they determined up to an additive 2, we obtain the exact maximum for n ∈ 𝒳m and for m ∈ {6, 10, 14}. For m = 10 it is less than V(10, n) exactly when n ≡ 2 (mod 3) and n ∉ 𝒳10, so on these infinitely many boards every cycle with the most turns skips lattice points; for m = 14 this happens only on [20] × [14]. We also disprove their conjecture that the error depends eventually only on n − m modulo 6. All proofs are by hand, and the finite verifications they use are printed in full. 2020 Mathematics Subject Classification: Primary 05B50; Secondary 05C38, 05C45, 52C05. Files: the paper (PDF) and a reproduction archive with programs and their stored outputs that independently confirm the exact values, the constructions, the printed certificates and the figures of the paper, including a checker that re-verifies from the PDF every construction datum printed in it; no proof depends on these programs. See README.md in the archive. Assisted by AI.