== 1. Berge-Hamiltonicity (the looser reading of 'Hamiltonian cycle') ==
H_3: Hamiltonian Berge cycle found: True; order (0, 1, 2, 3, 4, 5, 6)
example_k3_n9.txt: Hamiltonian Berge cycle found: True
example_k5_n10.txt: Hamiltonian Berge cycle found: True
   => only the tight reading (the one both OWR sources define) is refuted; a loose Hamiltonian cycle in a
      3-graph needs n even, so under the loose reading every 7-vertex 3-graph is a degenerate 'counterexample'.

== 2. forced walks in D' are the six vertex-avoiding tight 5-cycles ==
all 60 ordered starts: walk uses 5 points, and its 5 cyclic blocks = the 5 faces avoiding the 6th point: True

== 3. apex reduction and the path version ==
D' on 6 points: min pair-degree 2; tight Hamiltonian path: False   (RRS path threshold ceil(n/2)-1 = 2 at n = 6)
apex + G has a THC <=> G has a tight Hamiltonian path: confirmed on 200 random 3-graphs G (m = 5..7)
DP cross-check on H_3: THC exists = False

== 4. Katona-Kierstead sharpness construction H0 at small n (claimed only for n > k^2) ==
k=3 n= 5: delta_(k-1)(H0) = 1 (floor((n-k+1)/2) = 1); tight Hamiltonian cycle: True   (n > k^2: False)  [0.0s]
k=3 n= 6: delta_(k-1)(H0) = 2 (floor((n-k+1)/2) = 2); tight Hamiltonian cycle: False   (n > k^2: False)  [0.0s]
k=3 n= 7: delta_(k-1)(H0) = 2 (floor((n-k+1)/2) = 2); tight Hamiltonian cycle: False   (n > k^2: False)  [0.0s]
k=3 n= 8: delta_(k-1)(H0) = 3 (floor((n-k+1)/2) = 3); tight Hamiltonian cycle: False   (n > k^2: False)  [0.0s]
k=3 n= 9: delta_(k-1)(H0) = 3 (floor((n-k+1)/2) = 3); tight Hamiltonian cycle: False   (n > k^2: False)  [0.0s]
k=3 n=10: delta_(k-1)(H0) = 4 (floor((n-k+1)/2) = 4); tight Hamiltonian cycle: False   (n > k^2: True)  [0.0s]
k=3 n=11: delta_(k-1)(H0) = 4 (floor((n-k+1)/2) = 4); tight Hamiltonian cycle: False   (n > k^2: True)  [0.0s]
k=4 n= 6: delta_(k-1)(H0) = 1 (floor((n-k+1)/2) = 1); tight Hamiltonian cycle: True   (n > k^2: False)  [0.0s]
k=4 n= 7: delta_(k-1)(H0) = 2 (floor((n-k+1)/2) = 2); tight Hamiltonian cycle: False   (n > k^2: False)  [0.0s]
k=4 n= 8: delta_(k-1)(H0) = 2 (floor((n-k+1)/2) = 2); tight Hamiltonian cycle: False   (n > k^2: False)  [0.0s]
k=4 n= 9: delta_(k-1)(H0) = 3 (floor((n-k+1)/2) = 3); tight Hamiltonian cycle: False   (n > k^2: False)  [0.0s]
k=4 n=10: delta_(k-1)(H0) = 3 (floor((n-k+1)/2) = 3); tight Hamiltonian cycle: False   (n > k^2: False)  [0.0s]
k=4 n=11: delta_(k-1)(H0) = 4 (floor((n-k+1)/2) = 4); tight Hamiltonian cycle: False   (n > k^2: False)  [0.0s]
k=5 n= 7: delta_(k-1)(H0) = 1 (floor((n-k+1)/2) = 1); tight Hamiltonian cycle: True   (n > k^2: False)  [0.0s]
k=5 n= 8: delta_(k-1)(H0) = 2 (floor((n-k+1)/2) = 2); tight Hamiltonian cycle: True   (n > k^2: False)  [0.0s]
k=5 n= 9: delta_(k-1)(H0) = 2 (floor((n-k+1)/2) = 2); tight Hamiltonian cycle: True   (n > k^2: False)  [0.0s]
k=5 n=10: delta_(k-1)(H0) = 3 (floor((n-k+1)/2) = 3); tight Hamiltonian cycle: False   (n > k^2: False)  [0.1s]

== 5. typo reading of the source's cycle definition ('for i = l-k+2, ..., k' instead of '..., l') ==
H_3 has a tight Hamiltonian PATH: True  -> under the (typo) path reading H_3 is no
   counterexample; the intended definition is the cyclic one (explicit in OWR 48/2006 and in Liu-Liu 2021).
