Four-layer starters for the congruence class r ≡ 3 mod 4

Determine whether a four-layer balanced Hamilton starter exists for the directed circulant Cay(ℤ_{4r},{1,…,r}) when r ≡ 3 mod 4 by developing a construction different from the ABAB method, or establish a genuine structural obstruction in this congruence class.

Background

The paper constructs an explicit four-layer balanced Hamilton starter, together with a compatible step-2 deletion chain, for r ≡ 1 mod 4 and r ≥ 9. The construction uses an ABAB step word derived from a directed R*-terrace and relies on congruences for S = r(r+1)/2 that hold in the r ≡ 1 mod 4 case.

For r ≡ 3 mod 4, the authors explain that the ABAB construction fails because S ≡ 0 mod 2r and 2S ≡ 0 mod 4r, rather than the congruences needed for the complementary-prefix lift, Hamiltonicity, and four-layer balance. The unresolved issue is whether this failure is specific to the construction or reflects a genuine nonexistence phenomenon.