Verification of the "Ordered multiplicity inverse eigenvalue problem for 6 graphs"

1643 days ago by admin

### Load some basic files and create a few useful lists ### URL='https://raw.githubusercontent.com/jephianlin/mr_JG/master/' files=['Zq_c.pyx','Zq.py','zero_forcing_64.pyx','zero_forcing_wavefront.pyx','minrank.py', 'inertia.py'] # zero forcing and positive semidefinite zero forcing for f in files: load(URL+f) q=[3, 4, 4, 5, 5, 6, 3, 4, 4, 4, 3, 5, 4, 3, 4, 5, 4, 4, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] # minimum number of distinct eigenvalues (see Table 3) from sage.graphs.graph import Graph import networkx.generators.atlas G = [Graph(i) for i in networkx.generators.atlas.graph_atlas_g()] # graphs in order corresponding to "Atlas of Graphs" lookup = [G[i].canonical_label().graph6_string() for i in range(len(G))] # convenient labeling to_do = [77, 78, 79, 80, 81, 83, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208] # these are the connected graphs from the Atlas of Graphs with 6 vertices 
       
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### the complete solution to the inverse eigenvalue problem on 5 graphs ### we will use this to establish some lists for graphs with 6 vertices fives = {} fives[31] = [(1,1,1,1,1)] fives[29] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,3,1)] fives[30] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1)] fives[35] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1)] fives[36] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1)] fives[34] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1)] fives[38] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1)] fives[37] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2)] fives[40] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2)] fives[41] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2)] fives[42] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(3,1,1),(1,3,1),(1,1,3)] fives[43] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2)] fives[47] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2)] fives[44] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1)] fives[46] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1)] fives[45] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1)] fives[48] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1),(2,3),(3,2)] fives[49] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1),(2,3),(3,2)] fives[50] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1),(2,3),(3,2)] fives[51] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1),(2,3),(3,2)] fives[52] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1),(2,3),(3,2),(1,4),(4,1)] 
       
### Build up our dictionary to keep track of what to do ### sixes = {} for i in to_do: sixes[i] = {} for C in Compositions(6): if len(C) > 1: sixes[i][tuple(C)] = 0 # -1 = impossible; 0 = not determined; 1 = possible 
       
### Rule out using Z, Z_+, q (Section 4.1) ### count = 0 for i in to_do: qq = q[count] Z = zero_forcing_set_wavefront(G[i])[0] ZZ = Zplus(G[i]) for C in sixes[i]: if len(C) < qq: sixes[i][C] = -1 if max(C) > Z: sixes[i][C] = -1 if C[0] > ZZ or C[-1] > ZZ: sixes[i][C] = -1 count += 1 
       
### Use Perron-Frobenius to rule out odd-unicylic graphs (Lemma 7) ### for i in to_do: if G[i].size() == 6 and G[i].girth()%2: for C in sixes[i]: if sixes[i][C]==0 and min([C[0],C[-1]])>1: sixes[i][C] = -1 
       
### Rule out the 3-sun, generalized star consecutive eigenvalues (Lemma 8) ### for i in [77,78,94]: for C in [(1,1,2,2),(1,2,2,1),(2,2,1,1)]: sixes[i][C] = -1 
       
### Rule in using known IEGP for graphs on <=5 vertices (Theorem 1 and Theorem 2) ### refine = [[],[],[],[],[],[],[],[],[]] for i in to_do: if G[i].subgraph_search_count(G[52]): refine[0].append(i) elif G[i].subgraph_search_count(G[48]): refine[1].append(i) elif G[i].subgraph_search_count(G[18]): refine[2].append(i) elif G[i].subgraph_search_count(G[44]): refine[3].append(i) elif G[i].subgraph_search_count(G[16]) or G[i].subgraph_search_count(G[42]) or G[i].subgraph_search_count(G[106]): refine[4].append(i) elif G[i].subgraph_search_count(G[38]) or G[i].subgraph_search_count(G[34]): refine[5].append(i) elif G[i].subgraph_search_count(G[7]): refine[6].append(i) elif G[i].subgraph_search_count(G[13]): refine[7].append(i) elif G[i].subgraph_search_count(G[1]): refine[8].append(i) inherit = [0,0,0,0,0,0,0,0,0] inherit[0]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1),(2,2,1,1),(1,1,1,3),(1,1,3,1),(1,3,1,1),(3,1,1,1),(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1),(1,1,4),(1,4,1),(4,1,1)] inherit[1]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1),(2,2,1,1),(1,1,1,3),(1,1,3,1),(1,3,1,1),(3,1,1,1),(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)] inherit[2]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1),(2,2,1,1),(1,1,1,3),(1,1,3,1),(1,3,1,1),(3,1,1,1)] inherit[3]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1),(2,2,1,1),(1,1,3,1),(1,3,1,1)] inherit[4]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1),(2,2,1,1)] inherit[5]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,2,1),(2,2,1,1)] inherit[6]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1)] inherit[7]=[(1,1,1,1,1,1),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1)] inherit[8]=[(1,1,1,1,1,1)] for i in range(9): for j in refine[i]: for C in inherit[i]: sixes[j][C] = 1 
       
### Now to run twinning code (Section 3; Corollary 4) ### twinning_up = {} for i in fives: twinning_up[i] = {} for j in range(5): H = copy(G[i]) H.add_edge([j,5]) for k in H.neighbors(j): if not k==5: H.add_edge([k,5]) new_graph = lookup.index(H.canonical_label().graph6_string()) twinning_up[i][new_graph] = set([]) for C in fives[i]: D = list(C) D[0] += 1 if sixes[new_graph][tuple(D)] == 0: twinning_up[i][new_graph].add(tuple(D)) D = list(C) D[-1] += 1 if sixes[new_graph][tuple(D)] == 0: twinning_up[i][new_graph].add(tuple(D)) for i in twinning_up: for j in twinning_up[i]: for C in twinning_up[i][j]: if sixes[j][C] == 0: sixes[j][C] = 1 
       
### Strong twins (Section 3; Theorem 3) ### strong_twinning = [(29, 77, 4), (30, 78, 3), (30, 100, 3), (30, 114, 3), (34, 133, 3), (35, 119, 3), (35, 134, 3), (36, 142, 3), (37, 126, 3), (40, 144, 3), (42, 165, 4), (44, 146, 4), (45, 165, 4), (45, 191, 4), (46, 161, 4), (48, 190, 4), (48, 194, 4), (49, 195, 4), (49, 200, 4), (52, 208, 5)] for D in strong_twinning: for C in fives[D[0]]: for i in range(len(C)): if C[i] + 1 == D[2]: CC = list(copy(C)) CC[i] += 1 CC = tuple(CC) if sixes[D[1]][CC] == 0: sixes[D[1]][CC] = 1 
       
### The graph G96 has 222 and SSP (Proposition 20) ### for i in to_do: if G[i].subgraph_search_count(G[96]) and sixes[i][(2,2,2)] == 0: sixes[i][(2,2,2)] = 1 
       
### The following are ruled out by various proofs in the paper (Section 4.3) ### out = [ [138,[(1,3,2),(2,3,1)]], [125,[(1,3,2),(2,3,1)]], [121,[(1,3,2),(2,3,1)]], [117,[(1,3,2),(2,3,1)]], [133,[(3,1,2),(2,1,3)]], [117,[(3,1,2),(2,1,3)]], [153,[(3,1,2),(2,1,3)]] ] for x in out: for y in x[1]: sixes[x[0]][y] = -1 
       
### Include (3,3) by use of q-group ### for i in to_do: if sixes[i][(3,3)] == 0 and sixes[i][(4,2)] == -1: sixes[i][(3,3)] = 1 ### Construction of (3,3) and (4,2) for G204 ### sixes[204][(3,3)] = 1 sixes[204][(2,4)] = 1 sixes[204][(4,2)] = 1 ### all the spectrally arbitrary things that come with (3,3) and (4,2) (Section 5.4) ### for i in to_do: if sixes[i][(3,3)] == 1: for C in [(3,1,1,1),(3,2,1),(3,1,2),(2,1,3),(1,2,3),(1,1,1,3)]: sixes[i][C] = 1 for i in to_do: if sixes[i][(4,2)] == 1: for C in [(4,1,1), (1,1,4)]: sixes[i][C] = 1 
       
### documented in paper; spectrally arbitrary ### sixes[99][(2,2,2)] = 1 sixes[77][(1,1,3,1)] = 1 sixes[77][(1,3,1,1)] = 1 sixes[79][(1,2,2,1)] = 1 sixes[105][(1,1,1,1,2)] = 1 sixes[105][(1,1,1,2,1)] = 1 sixes[105][(1,1,2,1,1)] = 1 sixes[105][(1,2,1,1,1)] = 1 sixes[105][(2,1,1,1,1)] = 1 sixes[105][(1,1,2,2)] = 1 sixes[105][(1,2,1,2)] = -1 sixes[105][(2,1,1,2)] = 1 sixes[105][(1,2,2,1)] = 1 sixes[105][(2,1,2,1)] = -1 sixes[105][(2,2,1,1)] = 1 sixes[105][(2,2,2)] = 1 sixes[151][(2,1,3)] = 1 sixes[151][(3,1,2)] = 1 sixes[151][(3,1,1,1)] = 1 sixes[151][(1,1,1,3)] = 1 sixes[171][(2,1,3)] = 1 sixes[171][(3,1,2)] = 1 sixes[171][(3,1,1,1)] = 1 sixes[171][(1,1,1,3)] = 1 sixes[187][(2,1,3)] = 1 sixes[187][(3,1,2)] = 1 sixes[187][(3,1,1,1)] = 1 sixes[187][(1,1,1,3)] = 1 sixes[163][(3,1,2)] = 1 sixes[163][(2,1,3)] = 1 sixes[163][(3,1,1,1)] = 1 sixes[163][(1,1,1,3)] = 1 
       
### G175 not spectrally arbitrary for (1,4,1) (Section 6) ### sixes[175][(1,4,1)] = 1 ### Using G189 for (1,4,1) having SSP, spectrally arbitrary ### sixes[189][(1,4,1)] for i in to_do: if sixes[i][(1,4,1)] == 0 and G[i].subgraph_search_count(G[189]): sixes[i][(1,4,1)] = 1 ### Construction for G204 ### sixes[204][(1,4,1)] = 1 
       
five_lookup = {} for i in fives: five_lookup[G[i].canonical_label().graph6_string()] = i def list_minors(i): H = copy(G[i]) minors = set([]) for e in H.edges(): K = copy(H) K.delete_vertex(e[1]) for v in H.neighbors(e[1]): if not v == e[0]: K.add_edge([e[0],v]) minors.add(five_lookup[K.canonical_label().graph6_string()]) return minors 
       
for i in to_do: S = list_minors(i) if sixes[i][(1,3,1,1)] == 0 and ((44 in S) or (46 in S) or (45 in S) or (48 in S) or (49 in S) or (50 in S) or (51 in S) or (52 in S)): print (1,3,1,1), i, S elif sixes[i][(1,3,1,1)] == 0: print (1,3,1,1), i, "exceptional" 
       
(1, 3, 1, 1) 92 exceptional
(1, 3, 1, 1) 117 exceptional
(1, 3, 1, 1) 129 set([43, 44])
(1, 3, 1, 1) 130 exceptional
(1, 3, 1, 1) 145 set([34, 43, 46])
(1, 3, 1, 1) 150 exceptional
(1, 3, 1, 1) 151 set([48, 43, 44, 47])
(1, 3, 1, 1) 153 set([48, 43, 38])
(1, 3, 1, 1) 163 exceptional
(1, 3, 1, 1) 171 set([48, 50, 43, 47])
(1, 3, 1, 1) 174 set([48, 50])
(1, 3, 1, 1) 187 set([50, 47])
(1, 3, 1, 1) 92 exceptional
(1, 3, 1, 1) 117 exceptional
(1, 3, 1, 1) 129 set([43, 44])
(1, 3, 1, 1) 130 exceptional
(1, 3, 1, 1) 145 set([34, 43, 46])
(1, 3, 1, 1) 150 exceptional
(1, 3, 1, 1) 151 set([48, 43, 44, 47])
(1, 3, 1, 1) 153 set([48, 43, 38])
(1, 3, 1, 1) 163 exceptional
(1, 3, 1, 1) 171 set([48, 50, 43, 47])
(1, 3, 1, 1) 174 set([48, 50])
(1, 3, 1, 1) 187 set([50, 47])
for i in to_do: S = list_minors(i) if sixes[i][(2,3,1)] == 0 and ((45 in S) or (48 in S) or (49 in S) or (50 in S) or (51 in S) or (52 in S)): print (2,3,1), i, S elif sixes[i][(2,3,1)] == 0: print (2,3,1), i, "exceptional" 
       
(2, 3, 1) 114 exceptional
(2, 3, 1) 129 exceptional
(2, 3, 1) 135 exceptional
(2, 3, 1) 145 exceptional
(2, 3, 1) 146 exceptional
(2, 3, 1) 149 exceptional
(2, 3, 1) 150 exceptional
(2, 3, 1) 151 set([48, 43, 44, 47])
(2, 3, 1) 153 set([48, 43, 38])
(2, 3, 1) 154 set([48, 47])
(2, 3, 1) 162 exceptional
(2, 3, 1) 163 exceptional
(2, 3, 1) 169 set([49, 42, 43])
(2, 3, 1) 171 set([48, 50, 43, 47])
(2, 3, 1) 174 set([48, 50])
(2, 3, 1) 175 set([50])
(2, 3, 1) 187 set([50, 47])
(2, 3, 1) 114 exceptional
(2, 3, 1) 129 exceptional
(2, 3, 1) 135 exceptional
(2, 3, 1) 145 exceptional
(2, 3, 1) 146 exceptional
(2, 3, 1) 149 exceptional
(2, 3, 1) 150 exceptional
(2, 3, 1) 151 set([48, 43, 44, 47])
(2, 3, 1) 153 set([48, 43, 38])
(2, 3, 1) 154 set([48, 47])
(2, 3, 1) 162 exceptional
(2, 3, 1) 163 exceptional
(2, 3, 1) 169 set([49, 42, 43])
(2, 3, 1) 171 set([48, 50, 43, 47])
(2, 3, 1) 174 set([48, 50])
(2, 3, 1) 175 set([50])
(2, 3, 1) 187 set([50, 47])
### Using minors ### for i in [100]: sixes[i][(1,2,2,1)] = 1 for i in [129,145,153,151,171,174,187]: sixes[i][(1,3,1,1)] = 1 sixes[i][(1,1,3,1)] = 1 for i in [151,153,154,169,171,174,175,187]: sixes[i][(3,2,1)] = 1 sixes[i][(1,2,3)] = 1 sixes[i][(2,3,1)] = 1 sixes[i][(1,3,2)] = 1 
       
### Using SMP/SSP ### for i in sixes: if (sixes[i][(2,2,2)]==0) and G[i].subgraph_search_count(G[105]): sixes[i][(2,2,2)] = 1 sixes[i][(2,1,1,2)] = 1 for i in sixes: if (sixes[i][(2,2,2)]==0) and G[i].subgraph_search_count(G[125]): sixes[i][(2,2,2)] = 1 for i in sixes: if (sixes[i][(1,3,2)]==0) and G[i].subgraph_search_count(G[129]): sixes[i][(1,3,2)] = 1 sixes[i][(2,3,1)] = 1 
       
### sporadic cases ### sixes[92][(1,1,3,1)] = 1 sixes[92][(1,3,1,1)] = 1 sixes[94][(1,2,1,2)] = 1 sixes[94][(2,1,2,1)] = 1 sixes[114][(1,3,2)] = 1 sixes[114][(2,3,1)] = 1 sixes[115][(2,2,2)] = 1 sixes[117][(1,1,3,1)] = 1 sixes[117][(1,3,1,1)] = 1 sixes[130][(1,1,3,1)] = 1 sixes[130][(1,3,1,1)] = 1 sixes[135][(1,3,2)] = 1 sixes[135][(2,3,1)] = 1 sixes[146][(1,3,2)] = 1 sixes[146][(2,3,1)] = 1 sixes[150][(1,1,3,1)] = 1 sixes[150][(1,3,1,1)] = 1 sixes[150][(2,3,1)] = 1 sixes[150][(1,3,2)] = 1 sixes[163][(1,1,3,1)] = 1 sixes[163][(1,3,1,1)] = 1 sixes[163][(2,3,1)] = 1 sixes[163][(1,3,2)] = 1 sixes[163][(1,2,3)] = 1 sixes[163][(3,2,1)] = 1 ### cite q for G129 ### sixes[129][(2,2,2)] = 1 
       
left = 0 for i in to_do: for C in sixes[i]: if sixes[i][C] == 0: print i,C left += 1 print left ### Note output is "0" if all cases are accounted for 
       
0
0
# Equivalence classes of graphs groupings = [] for i in sixes: for g in groupings: if sixes[i]==sixes[g[0]]: g.append(i) break else: groupings.append([i]) for g in groupings: print g print str(len(groupings))+" equicavelence classes of graphs" 
       
[77]
[78]
[79]
[80, 81]
[83]
[92]
[93, 95, 104]
[94]
[96, 99, 111, 115, 118, 127, 128, 136, 137, 147, 148, 152, 164, 167]
[97, 102]
[98, 103, 112, 113, 120, 122, 123, 124, 139]
[100]
[105]
[114, 129, 135, 145, 149, 162]
[117]
[119, 130, 134, 142]
[121, 125, 138]
[126, 140, 141, 143, 144, 150, 151, 156, 157, 158, 159, 160, 163, 166,
169, 170, 171, 172, 173, 177, 178, 179, 180, 182, 183, 184, 185, 187,
193]
[133, 153]
[146, 161]
[154, 168, 174, 181, 186, 188, 192, 196, 198, 202]
[165, 191]
[175, 197, 201]
[189]
[190, 194, 195, 199, 200, 203, 204, 205, 206, 207]
[208]
26 equicavelence classes of graphs
[77]
[78]
[79]
[80, 81]
[83]
[92]
[93, 95, 104]
[94]
[96, 99, 111, 115, 118, 127, 128, 136, 137, 147, 148, 152, 164, 167]
[97, 102]
[98, 103, 112, 113, 120, 122, 123, 124, 139]
[100]
[105]
[114, 129, 135, 145, 149, 162]
[117]
[119, 130, 134, 142]
[121, 125, 138]
[126, 140, 141, 143, 144, 150, 151, 156, 157, 158, 159, 160, 163, 166, 169, 170, 171, 172, 173, 177, 178, 179, 180, 182, 183, 184, 185, 187, 193]
[133, 153]
[146, 161]
[154, 168, 174, 181, 186, 188, 192, 196, 198, 202]
[165, 191]
[175, 197, 201]
[189]
[190, 194, 195, 199, 200, 203, 204, 205, 206, 207]
[208]
26 equicavelence classes of graphs
sixes[152] # Example of what is produced for a particular graph 
       
{(1, 1, 1, 1, 1, 1): 1,
 (1, 1, 1, 1, 2): 1,
 (1, 1, 1, 2, 1): 1,
 (1, 1, 1, 3): -1,
 (1, 1, 2, 1, 1): 1,
 (1, 1, 2, 2): 1,
 (1, 1, 3, 1): -1,
 (1, 1, 4): -1,
 (1, 2, 1, 1, 1): 1,
 (1, 2, 1, 2): 1,
 (1, 2, 2, 1): 1,
 (1, 2, 3): -1,
 (1, 3, 1, 1): -1,
 (1, 3, 2): -1,
 (1, 4, 1): -1,
 (1, 5): -1,
 (2, 1, 1, 1, 1): 1,
 (2, 1, 1, 2): 1,
 (2, 1, 2, 1): 1,
 (2, 1, 3): -1,
 (2, 2, 1, 1): 1,
 (2, 2, 2): 1,
 (2, 3, 1): -1,
 (2, 4): -1,
 (3, 1, 1, 1): -1,
 (3, 1, 2): -1,
 (3, 2, 1): -1,
 (3, 3): -1,
 (4, 1, 1): -1,
 (4, 2): -1,
 (5, 1): -1}
{(1, 1, 1, 1, 1, 1): 1,
 (1, 1, 1, 1, 2): 1,
 (1, 1, 1, 2, 1): 1,
 (1, 1, 1, 3): -1,
 (1, 1, 2, 1, 1): 1,
 (1, 1, 2, 2): 1,
 (1, 1, 3, 1): -1,
 (1, 1, 4): -1,
 (1, 2, 1, 1, 1): 1,
 (1, 2, 1, 2): 1,
 (1, 2, 2, 1): 1,
 (1, 2, 3): -1,
 (1, 3, 1, 1): -1,
 (1, 3, 2): -1,
 (1, 4, 1): -1,
 (1, 5): -1,
 (2, 1, 1, 1, 1): 1,
 (2, 1, 1, 2): 1,
 (2, 1, 2, 1): 1,
 (2, 1, 3): -1,
 (2, 2, 1, 1): 1,
 (2, 2, 2): 1,
 (2, 3, 1): -1,
 (2, 4): -1,
 (3, 1, 1, 1): -1,
 (3, 1, 2): -1,
 (3, 2, 1): -1,
 (3, 3): -1,
 (4, 1, 1): -1,
 (4, 2): -1,
 (5, 1): -1}
check = [(1,1,1,1,1,1), (1,1,1,1,2), (1,1,1,2,1), (1,1,2,1,1), (1,1,2,2), (1,2,1,2), (1,2,2,1), (2,1,1,2), (2,2,2), (1,1,1,3), (1,1,3,1), (1,2,3), (1,3,2), (2,1,3), (3,3), (1,1,4), (1,4,1), (2,4), (1,5)] write = ["111111, ", "11112, 21111, ", "11121, 12111, ", "11211, ", "1122, 2211, ", "1212, 2121, ", "1221, ", "2112, ", "222, ", "1113, 3111, ", "1131, 1311, ", "123, 321, ", "132, 231, ", "213, 312, ", "33, ", "114, 411, ", "141, ", "24, 42, ", "15, 51, "] 
       
# Use the data to indicate for each graph which ordered multiplicity lists are attainable (Appendix B) for i in to_do: S="" for C in range(19): if sixes[i][check[C]] == 1: S+=write[C] S = S[:-2] print "For graph "+str(i)+": {"+S+"}" 
       
For graph 77: {111111, 11121, 12111, 11211, 1131, 1311, 141}
For graph 78: {111111, 11121, 12111, 11211, 1131, 1311}
For graph 79: {111111, 11121, 12111, 11211, 1221}
For graph 80: {111111, 11121, 12111, 11211}
For graph 81: {111111, 11121, 12111, 11211}
For graph 83: {111111}
For graph 92: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 1131, 1311, 132, 231}
For graph 93: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221}
For graph 94: {111111, 11112, 21111, 11121, 12111, 11211, 1212, 2121}
For graph 95: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221}
For graph 96: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 97: {111111, 11112, 21111, 11121, 12111, 11211}
For graph 98: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 99: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 100: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 1131, 1311}
For graph 102: {111111, 11112, 21111, 11121, 12111, 11211}
For graph 103: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 104: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221}
For graph 105: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1221, 2112, 222}
For graph 111: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 112: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 113: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 114: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 115: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 117: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321}
For graph 118: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 119: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 120: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 121: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 122: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 123: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 124: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 125: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 126: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 127: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 128: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 129: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 130: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 133: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231}
For graph 134: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 135: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 136: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 137: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 138: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 139: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 140: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 141: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 142: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 143: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 144: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 145: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 146: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231, 141}
For graph 147: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 148: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 149: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 150: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 151: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 152: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 153: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231}
For graph 154: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 156: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 157: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 158: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 159: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 160: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 161: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231, 141}
For graph 162: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 163: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 164: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 165: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 114, 411, 141}
For graph 166: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 167: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 168: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 169: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 170: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 171: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 172: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 173: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 174: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 175: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 141}
For graph 177: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 178: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 179: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 180: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 181: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 182: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 183: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 184: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 185: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 186: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 187: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 188: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 189: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 141}
For graph 190: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 191: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 114, 411, 141}
For graph 192: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 193: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 194: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 195: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 196: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 197: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 141}
For graph 198: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 199: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 200: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 201: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 141}
For graph 202: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 203: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 204: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 205: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 206: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 207: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 208: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42, 15, 51}
For graph 77: {111111, 11121, 12111, 11211, 1131, 1311, 141}
For graph 78: {111111, 11121, 12111, 11211, 1131, 1311}
For graph 79: {111111, 11121, 12111, 11211, 1221}
For graph 80: {111111, 11121, 12111, 11211}
For graph 81: {111111, 11121, 12111, 11211}
For graph 83: {111111}
For graph 92: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 1131, 1311, 132, 231}
For graph 93: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221}
For graph 94: {111111, 11112, 21111, 11121, 12111, 11211, 1212, 2121}
For graph 95: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221}
For graph 96: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 97: {111111, 11112, 21111, 11121, 12111, 11211}
For graph 98: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 99: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 100: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 1131, 1311}
For graph 102: {111111, 11112, 21111, 11121, 12111, 11211}
For graph 103: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 104: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221}
For graph 105: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1221, 2112, 222}
For graph 111: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 112: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 113: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 114: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 115: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 117: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321}
For graph 118: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 119: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 120: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 121: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 122: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 123: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 124: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 125: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 126: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 127: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 128: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 129: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 130: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 133: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231}
For graph 134: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 135: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 136: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 137: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 138: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 139: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 140: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 141: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 142: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 143: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 144: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 145: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 146: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231, 141}
For graph 147: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 148: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 149: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 150: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 151: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 152: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 153: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231}
For graph 154: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 156: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 157: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 158: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 159: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 160: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 161: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231, 141}
For graph 162: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 163: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 164: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 165: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 114, 411, 141}
For graph 166: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 167: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 168: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 169: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 170: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 171: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 172: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 173: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 174: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 175: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 141}
For graph 177: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 178: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 179: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 180: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 181: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 182: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 183: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 184: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 185: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 186: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 187: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 188: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 189: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 141}
For graph 190: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 191: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 114, 411, 141}
For graph 192: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 193: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 194: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 195: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 196: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 197: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 141}
For graph 198: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 199: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 200: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 201: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 141}
For graph 202: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 203: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 204: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 205: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 206: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 207: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 208: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42, 15, 51}
# Use the data to indicate for each ordered multiplicity list which graphs attain it (Appendix A) for C in range(19): S="" for i in to_do: if sixes[i][check[C]] == 1: S+=str(i)+", " S = S[:-2] print "For list "+write[C][:-2]+": {"+S+"}" 
       
For list 111111: {77, 78, 79, 80, 81, 83, 92, 93, 94, 95, 96, 97, 98,
99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119,
120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135,
136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149,
150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164,
165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179,
180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193,
194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207,
208}
For list 11112, 21111: {92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103,
104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123,
124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139,
140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153,
154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168,
169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183,
184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197,
198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 11121, 12111: {77, 78, 79, 80, 81, 92, 93, 94, 95, 96, 97, 98,
99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119,
120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135,
136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149,
150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164,
165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179,
180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193,
194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207,
208}
For list 11211: {77, 78, 79, 80, 81, 92, 93, 94, 95, 96, 97, 98, 99,
100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120,
121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136,
137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150,
151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165,
166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180,
181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194,
195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1122, 2211: {92, 93, 95, 96, 98, 99, 100, 103, 104, 105, 111,
112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157,
158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171,
172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186,
187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200,
201, 202, 203, 204, 205, 206, 207, 208}
For list 1212, 2121: {92, 93, 94, 95, 96, 98, 99, 100, 103, 104, 111,
112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157,
158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171,
172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186,
187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200,
201, 202, 203, 204, 205, 206, 207, 208}
For list 1221: {79, 92, 93, 95, 96, 98, 99, 100, 103, 104, 105, 111,
112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157,
158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171,
172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186,
187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200,
201, 202, 203, 204, 205, 206, 207, 208}
For list 2112: {96, 98, 99, 103, 105, 111, 112, 113, 114, 115, 117, 118,
119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134,
135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148,
149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163,
164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178,
179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192,
193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206,
207, 208}
For list 222: {96, 99, 105, 111, 114, 115, 117, 118, 121, 125, 126, 127,
128, 129, 133, 135, 136, 137, 138, 140, 141, 143, 144, 145, 146, 147,
148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162,
163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177,
178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191,
192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205,
206, 207, 208}
For list 1113, 3111: {117, 119, 126, 130, 133, 134, 140, 141, 142, 143,
144, 150, 151, 153, 154, 156, 157, 158, 159, 160, 163, 165, 166, 168,
169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183,
184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197,
198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1131, 1311: {77, 78, 92, 100, 114, 117, 119, 121, 125, 126,
129, 130, 133, 134, 135, 138, 140, 141, 142, 143, 144, 145, 146, 149,
150, 151, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 165, 166,
168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182,
183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196,
197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 123, 321: {117, 126, 133, 140, 141, 143, 144, 150, 151, 153,
154, 156, 157, 158, 159, 160, 163, 165, 166, 168, 169, 170, 171, 172,
173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187,
188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201,
202, 203, 204, 205, 206, 207, 208}
For list 132, 231: {92, 114, 126, 129, 133, 135, 140, 141, 143, 144,
145, 146, 149, 150, 151, 153, 154, 156, 157, 158, 159, 160, 161, 162,
163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179,
180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193,
194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207,
208}
For list 213, 312: {126, 140, 141, 143, 144, 150, 151, 154, 156, 157,
158, 159, 160, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175,
177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190,
191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204,
205, 206, 207, 208}
For list 33: {154, 168, 174, 175, 181, 186, 188, 190, 192, 194, 195,
196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 114, 411: {165, 190, 191, 194, 195, 199, 200, 203, 204, 205,
206, 207, 208}
For list 141: {77, 146, 161, 165, 175, 189, 190, 191, 194, 195, 197,
199, 200, 201, 203, 204, 205, 206, 207, 208}
For list 24, 42: {190, 194, 195, 199, 200, 203, 204, 205, 206, 207, 208}
For list 15, 51: {208}
For list 111111: {77, 78, 79, 80, 81, 83, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 11112, 21111: {92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 11121, 12111: {77, 78, 79, 80, 81, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 11211: {77, 78, 79, 80, 81, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1122, 2211: {92, 93, 95, 96, 98, 99, 100, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1212, 2121: {92, 93, 94, 95, 96, 98, 99, 100, 103, 104, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1221: {79, 92, 93, 95, 96, 98, 99, 100, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 2112: {96, 98, 99, 103, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 222: {96, 99, 105, 111, 114, 115, 117, 118, 121, 125, 126, 127, 128, 129, 133, 135, 136, 137, 138, 140, 141, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1113, 3111: {117, 119, 126, 130, 133, 134, 140, 141, 142, 143, 144, 150, 151, 153, 154, 156, 157, 158, 159, 160, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1131, 1311: {77, 78, 92, 100, 114, 117, 119, 121, 125, 126, 129, 130, 133, 134, 135, 138, 140, 141, 142, 143, 144, 145, 146, 149, 150, 151, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 123, 321: {117, 126, 133, 140, 141, 143, 144, 150, 151, 153, 154, 156, 157, 158, 159, 160, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 132, 231: {92, 114, 126, 129, 133, 135, 140, 141, 143, 144, 145, 146, 149, 150, 151, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 213, 312: {126, 140, 141, 143, 144, 150, 151, 154, 156, 157, 158, 159, 160, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 33: {154, 168, 174, 175, 181, 186, 188, 190, 192, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 114, 411: {165, 190, 191, 194, 195, 199, 200, 203, 204, 205, 206, 207, 208}
For list 141: {77, 146, 161, 165, 175, 189, 190, 191, 194, 195, 197, 199, 200, 201, 203, 204, 205, 206, 207, 208}
For list 24, 42: {190, 194, 195, 199, 200, 203, 204, 205, 206, 207, 208}
For list 15, 51: {208}