hemi-icosahedron D' (10 faces): [[1, 2, 3], [1, 2, 4], [1, 3, 5], [1, 4, 6], [1, 5, 6], [2, 3, 6], [2, 4, 5], [2, 5, 6], [3, 4, 5], [3, 4, 6]]
  D': every pair of W in exactly 2 blocks: True
  D: every pair of W in exactly 2 blocks: True
  (F1) T in D' <=> W - T in D  (so D and D' contain no complementary pair): True
  (F2) longest tight path in D' (x1..xm distinct, all x_i x_i+1 x_i+2 blocks): 5 vertices
  (F2) longest tight path in D (x1..xm distinct, all x_i x_i+1 x_i+2 blocks): 5 vertices

H_3: n=7, |E|=25, min codegree 3 (conjectured sufficient: 3); tight Hamiltonian cycles among all 360 cyclic orders: 0  [0.0s]
H_4: n=8, |E|=50, min codegree 3 (conjectured sufficient: 3); tight Hamiltonian cycles among all 2520 cyclic orders: 0  [0.0s]
H_5: n=9, |E|=96, min codegree 3 (conjectured sufficient: 3); tight Hamiltonian cycles among all 20160 cyclic orders: 0  [0.0s]
case analysis: all 83 gap vectors with 1..3 apexes force a complementary pair or 4 consecutive triples

example_k3_n9.txt: n=9, k=3, |E|=50, min codegree 4; floor((n-k+3)/2) = 4, (n-k+3)/2 = 4.5; tight Hamiltonian cycle found: False  [0.0s]
example_k5_n10.txt: n=10, k=5, |E|=191, min codegree 4; floor((n-k+3)/2) = 4, (n-k+3)/2 = 4.0; tight Hamiltonian cycle found: False  [0.3s]
  (the 5-graph has min codegree 4 = (n-k+3)/2 exactly: it refutes the statement with or without the floor,
   and also the exact large-n threshold ceil((n-k+2)/2) = 4 at n = 10.)

k=3, n=4: 3-graphs with min codegree >= 2: 1; without tight Hamiltonian cycle: 0
k=3, n=5: 3-graphs with min codegree >= 2: 26; without tight Hamiltonian cycle: 0
k=3, n=6: 3-graphs with min codegree >= 3: 271; without tight Hamiltonian cycle: 0
[total 3.2s]
