Part A: the graphs F_c^(k) and their blow-ups
  k=3 c=1 a= 1: n=   9 w=   7 n-w=  2 (n-1)-(2k-1)(n-w)=-2  (M) holds; with K_1: n=  10 (M) holds
  k=3 c=2 a= 3: n=  20 w=  16 n-w=  4 (n-1)-(2k-1)(n-w)=-1  (M) holds; with K_1: n=  21 (M) holds
  k=3 c=3 a= 6: n=  36 w=  29 n-w=  7 (n-1)-(2k-1)(n-w)= 0  (M) holds; with K_1: n=  37 (M) FAILS
        blow-up t=2: n=  72 w=  58 (n-1)-(2k-1)(n-w)= 1
        blow-up t=3: n= 108 w=  87 (n-1)-(2k-1)(n-w)= 2
  k=3 c=4 a=10: n=  57 w=  46 n-w= 11 (n-1)-(2k-1)(n-w)= 1  (M) FAILS; with K_1: n=  58 (M) FAILS
  k=3 c=5 a=15: n=  83 w=  67 n-w= 16 (n-1)-(2k-1)(n-w)= 2  (M) FAILS; with K_1: n=  84 (M) FAILS
        blow-up t=2: n= 166 w= 134 (n-1)-(2k-1)(n-w)= 5
        blow-up t=3: n= 249 w= 201 (n-1)-(2k-1)(n-w)= 8
  k=3 c=6 a=21: n= 114 w=  92 n-w= 22 (n-1)-(2k-1)(n-w)= 3  (M) FAILS; with K_1: n= 115 (M) FAILS
  k=4 c=1 a= 1: n=  13 w=  11 n-w=  2 (n-1)-(2k-1)(n-w)=-2  (M) holds; with K_1: n=  14 (M) holds
  k=4 c=2 a= 3: n=  28 w=  24 n-w=  4 (n-1)-(2k-1)(n-w)=-1  (M) holds; with K_1: n=  29 (M) holds
  k=4 c=3 a= 6: n=  50 w=  43 n-w=  7 (n-1)-(2k-1)(n-w)= 0  (M) holds; with K_1: n=  51 (M) FAILS
        blow-up t=2: n= 100 w=  86 (n-1)-(2k-1)(n-w)= 1
        blow-up t=3: n= 150 w= 129 (n-1)-(2k-1)(n-w)= 2
  k=4 c=4 a=10: n=  79 w=  68 n-w= 11 (n-1)-(2k-1)(n-w)= 1  (M) FAILS; with K_1: n=  80 (M) FAILS
  k=4 c=5 a=15: n= 115 w=  99 n-w= 16 (n-1)-(2k-1)(n-w)= 2  (M) FAILS; with K_1: n= 116 (M) FAILS
        blow-up t=2: n= 230 w= 198 (n-1)-(2k-1)(n-w)= 5
        blow-up t=3: n= 345 w= 297 (n-1)-(2k-1)(n-w)= 8
  k=4 c=6 a=21: n= 158 w= 136 n-w= 22 (n-1)-(2k-1)(n-w)= 3  (M) FAILS; with K_1: n= 159 (M) FAILS
  k=5 c=1 a= 1: n=  17 w=  15 n-w=  2 (n-1)-(2k-1)(n-w)=-2  (M) holds; with K_1: n=  18 (M) holds
  k=5 c=2 a= 3: n=  36 w=  32 n-w=  4 (n-1)-(2k-1)(n-w)=-1  (M) holds; with K_1: n=  37 (M) holds
  k=5 c=3 a= 6: n=  64 w=  57 n-w=  7 (n-1)-(2k-1)(n-w)= 0  (M) holds; with K_1: n=  65 (M) FAILS
        blow-up t=2: n= 128 w= 114 (n-1)-(2k-1)(n-w)= 1
        blow-up t=3: n= 192 w= 171 (n-1)-(2k-1)(n-w)= 2
  k=5 c=4 a=10: n= 101 w=  90 n-w= 11 (n-1)-(2k-1)(n-w)= 1  (M) FAILS; with K_1: n= 102 (M) FAILS
  k=5 c=5 a=15: n= 147 w= 131 n-w= 16 (n-1)-(2k-1)(n-w)= 2  (M) FAILS; with K_1: n= 148 (M) FAILS
        blow-up t=2: n= 294 w= 262 (n-1)-(2k-1)(n-w)= 5
        blow-up t=3: n= 441 w= 393 (n-1)-(2k-1)(n-w)= 8
  k=5 c=6 a=21: n= 202 w= 180 n-w= 22 (n-1)-(2k-1)(n-w)= 3  (M) FAILS; with K_1: n= 203 (M) FAILS
  k=6 c=1 a= 1: n=  21 w=  19 n-w=  2 (n-1)-(2k-1)(n-w)=-2  (M) holds; with K_1: n=  22 (M) holds
  k=6 c=2 a= 3: n=  44 w=  40 n-w=  4 (n-1)-(2k-1)(n-w)=-1  (M) holds; with K_1: n=  45 (M) holds
  k=6 c=3 a= 6: n=  78 w=  71 n-w=  7 (n-1)-(2k-1)(n-w)= 0  (M) holds; with K_1: n=  79 (M) FAILS
  k=6 c=4 a=10: n= 123 w= 112 n-w= 11 (n-1)-(2k-1)(n-w)= 1  (M) FAILS; with K_1: n= 124 (M) FAILS
  k=6 c=5 a=15: n= 179 w= 163 n-w= 16 (n-1)-(2k-1)(n-w)= 2  (M) FAILS; with K_1: n= 180 (M) FAILS
  k=6 c=6 a=21: n= 246 w= 224 n-w= 22 (n-1)-(2k-1)(n-w)= 3  (M) FAILS; with K_1: n= 247 (M) FAILS
  k=7 c=1 a= 1: n=  25 w=  23 n-w=  2 (n-1)-(2k-1)(n-w)=-2  (M) holds; with K_1: n=  26 (M) holds
  k=7 c=2 a= 3: n=  52 w=  48 n-w=  4 (n-1)-(2k-1)(n-w)=-1  (M) holds; with K_1: n=  53 (M) holds
  k=7 c=3 a= 6: n=  92 w=  85 n-w=  7 (n-1)-(2k-1)(n-w)= 0  (M) holds; with K_1: n=  93 (M) FAILS
  k=7 c=4 a=10: n= 145 w= 134 n-w= 11 (n-1)-(2k-1)(n-w)= 1  (M) FAILS; with K_1: n= 146 (M) FAILS
  k=7 c=5 a=15: n= 211 w= 195 n-w= 16 (n-1)-(2k-1)(n-w)= 2  (M) FAILS; with K_1: n= 212 (M) FAILS
  k=7 c=6 a=21: n= 290 w= 268 n-w= 22 (n-1)-(2k-1)(n-w)= 3  (M) FAILS; with K_1: n= 291 (M) FAILS
  Part A: all assertions passed

Part B: Lemma 2.1 checked for all 2 <= n <= 200, 1 <= w < n, 1 <= chi <= n

Part C: Proposition 5.1 checked on all 108622 spanning subgraphs of K_{a,b}, a >= b >= 1, a + b <= 8; sharpness example checked for n = 3p, p < 30

Part D(i): Theorem 5.3 verified by brute force on all 33866 labelled graphs with 2 <= n <= 6
Part D(ii): delta=0: pairs (n,k), n <= 60, k >= 3, NOT excluded by Theorem 5.3: [(37, 3), (42, 3), (47, 3), (51, 4), (52, 3), (57, 3), (58, 3), (58, 4)]
Part D(ii): delta=1: pairs (n,k), n <= 60, k >= 3, NOT excluded by Theorem 5.3: [(57, 3)]
Part D(ii): 890 certificates written to lagrange_certificates.txt
Part D(iii): for every 3 <= k <= 60 the least order not excluded by Theorem 5.3 is 14k-5 (all graphs) and 22k-9 (no isolated vertex)

Part E: n=100  B(n,3)/n = 0.19000   (3-sqrt5)/4 = 0.19098
Part E: n=200  B(n,3)/n = 0.19051   (3-sqrt5)/4 = 0.19098
Part E: n=300  B(n,3)/n = 0.19067   (3-sqrt5)/4 = 0.19098
        k=3: mu_k = 0.19098 <= liminf (n-w)/n <= 16/(32k-13) = 0.19277 < 1/(2k-1) = 0.20000
        k=4: mu_k = 0.13811 <= liminf (n-w)/n <= 16/(32k-13) = 0.13913 < 1/(2k-1) = 0.14286
        k=5: mu_k = 0.10819 <= liminf (n-w)/n <= 16/(32k-13) = 0.10884 < 1/(2k-1) = 0.11111
        k=6: mu_k = 0.08894 <= liminf (n-w)/n <= 16/(32k-13) = 0.08939 < 1/(2k-1) = 0.09091
        k=7: mu_k = 0.07550 <= liminf (n-w)/n <= 16/(32k-13) = 0.07583 < 1/(2k-1) = 0.07692

ALL CHECKS PASSED
