Values
=>
Cc0020;cc-rep
([],0)=>0
([],1)=>1
([],2)=>2
([(0,1)],2)=>2
([],3)=>3
([(1,2)],3)=>3
([(0,2),(1,2)],3)=>3
([(0,1),(0,2),(1,2)],3)=>6
([],4)=>4
([(2,3)],4)=>4
([(1,3),(2,3)],4)=>4
([(0,3),(1,3),(2,3)],4)=>6
([(0,3),(1,2)],4)=>5
([(0,3),(1,2),(2,3)],4)=>5
([(1,2),(1,3),(2,3)],4)=>6
([(0,3),(1,2),(1,3),(2,3)],4)=>7
([(0,2),(0,3),(1,2),(1,3)],4)=>6
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>10
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>18
([],5)=>5
([(3,4)],5)=>5
([(2,4),(3,4)],5)=>5
([(1,4),(2,4),(3,4)],5)=>6
([(0,4),(1,4),(2,4),(3,4)],5)=>7
([(1,4),(2,3)],5)=>5
([(1,4),(2,3),(3,4)],5)=>5
([(0,1),(2,4),(3,4)],5)=>6
([(2,3),(2,4),(3,4)],5)=>6
([(0,4),(1,4),(2,3),(3,4)],5)=>6
([(1,4),(2,3),(2,4),(3,4)],5)=>7
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>9
([(1,3),(1,4),(2,3),(2,4)],5)=>6
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)=>6
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>10
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>9
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>10
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)=>10
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>14
([(0,4),(1,3),(2,3),(2,4)],5)=>6
([(0,1),(2,3),(2,4),(3,4)],5)=>7
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>9
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)=>9
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)=>9
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>9
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)=>10
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)=>10
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>18
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>18
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>18
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)=>10
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)=>15
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>22
([],6)=>6
([(4,5)],6)=>6
([(3,5),(4,5)],6)=>6
([(2,5),(3,5),(4,5)],6)=>6
([(1,5),(2,5),(3,5),(4,5)],6)=>7
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>10
([(2,5),(3,4)],6)=>6
([(2,5),(3,4),(4,5)],6)=>6
([(1,2),(3,5),(4,5)],6)=>6
([(3,4),(3,5),(4,5)],6)=>6
([(1,5),(2,5),(3,4),(4,5)],6)=>6
([(0,1),(2,5),(3,5),(4,5)],6)=>7
([(2,5),(3,4),(3,5),(4,5)],6)=>7
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>7
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>9
([(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,5),(1,5),(2,4),(3,4)],6)=>7
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>6
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>7
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>10
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>9
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>8
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>7
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>10
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>10
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>14
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>14
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>18
([(0,5),(1,4),(2,3)],6)=>8
([(1,5),(2,4),(3,4),(3,5)],6)=>6
([(0,1),(2,5),(3,4),(4,5)],6)=>8
([(1,2),(3,4),(3,5),(4,5)],6)=>7
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>9
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>9
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>9
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>9
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>10
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>10
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>8
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>18
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>18
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>18
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>10
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>15
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)=>8
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>22
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)=>18
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)=>10
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>17
([],7)=>7
([(5,6)],7)=>7
([(4,6),(5,6)],7)=>7
([(3,6),(4,6),(5,6)],7)=>7
([(2,6),(3,6),(4,6),(5,6)],7)=>7
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>10
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>11
([(3,6),(4,5)],7)=>7
([(3,6),(4,5),(5,6)],7)=>7
([(2,3),(4,6),(5,6)],7)=>7
([(4,5),(4,6),(5,6)],7)=>7
([(2,6),(3,6),(4,5),(5,6)],7)=>7
([(1,2),(3,6),(4,6),(5,6)],7)=>7
([(3,6),(4,5),(4,6),(5,6)],7)=>7
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>7
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)=>9
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>9
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>10
([(3,5),(3,6),(4,5),(4,6)],7)=>7
([(1,6),(2,6),(3,5),(4,5)],7)=>7
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)=>7
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>7
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)=>8
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>10
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>9
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>8
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)=>7
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>9
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>10
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>9
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>10
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>14
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>14
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>18
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>18
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>21
([(1,6),(2,5),(3,4)],7)=>8
([(2,6),(3,5),(4,5),(4,6)],7)=>7
([(1,2),(3,6),(4,5),(5,6)],7)=>8
([(0,3),(1,2),(4,6),(5,6)],7)=>9
([(2,3),(4,5),(4,6),(5,6)],7)=>7
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)=>9
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>9
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)=>9
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>9
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>9
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>10
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)=>10
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)=>8
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)=>9
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>18
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>18
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>18
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)=>10
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)=>15
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)=>8
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>22
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)=>18
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)=>9
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)=>9
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)=>10
([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7)=>13
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>17
([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)=>11
([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)=>10
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)=>14
([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)=>11
([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)=>11
([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)=>10
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>18
([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>18
([(5,6),(5,7),(6,7)],8)=>8
([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>18
([(4,7),(5,6),(5,7),(6,7)],8)=>8
([(3,7),(4,7),(5,6),(5,7),(6,7)],8)=>9
([(6,7)],8)=>8
([(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>22
([(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>10
([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>10
([(5,7),(6,7)],8)=>8
([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>14
([(4,7),(5,7),(6,7)],8)=>8
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>18
([(3,7),(4,7),(5,7),(6,7)],8)=>8
([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>21
([(2,7),(3,7),(4,7),(5,7),(6,7)],8)=>10
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>26
([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)=>11
([],8)=>8
([(4,7),(5,6)],8)=>8
([(4,7),(5,6),(6,7)],8)=>8
([(3,4),(5,7),(6,7)],8)=>8
([(3,7),(4,7),(5,6),(6,7)],8)=>8
([(2,3),(4,7),(5,7),(6,7)],8)=>8
([(2,7),(3,7),(4,7),(5,6),(6,7)],8)=>8
([(1,2),(3,7),(4,7),(5,7),(6,7)],8)=>9
([(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)=>10
([(4,6),(4,7),(5,6),(5,7)],8)=>8
([(2,7),(3,7),(4,6),(5,6)],8)=>8
([(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>8
([(2,7),(3,7),(4,5),(5,6),(6,7)],8)=>8
([(1,7),(2,7),(3,7),(4,6),(5,6)],8)=>8
([(3,7),(4,6),(5,6),(5,7),(6,7)],8)=>9
([(2,7),(3,7),(4,6),(5,6),(6,7)],8)=>8
([(2,7),(3,7),(4,5),(4,6),(5,7),(6,7)],8)=>8
([(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)=>9
([(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)=>9
([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>10
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>14
([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>18
([(2,7),(3,6),(4,5)],8)=>8
([(3,7),(4,6),(5,6),(5,7)],8)=>8
([(2,3),(4,7),(5,6),(6,7)],8)=>8
([(1,4),(2,3),(5,7),(6,7)],8)=>9
([(3,4),(5,6),(5,7),(6,7)],8)=>8
([(3,6),(4,5),(4,7),(5,7),(6,7)],8)=>9
([(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)=>9
([(3,6),(3,7),(4,5),(4,7),(5,6)],8)=>9
([(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)=>9
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)=>9
([(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)=>10
([(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)=>10
([(2,7),(3,6),(4,5),(4,6),(5,7)],8)=>8
([(1,7),(2,6),(3,5),(4,5),(6,7)],8)=>9
([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)=>10
([(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)=>15
([(2,6),(2,7),(3,4),(3,5),(4,7),(5,6)],8)=>8
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)=>18
([(1,2),(3,6),(4,5),(5,7),(6,7)],8)=>9
([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)=>9
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)=>10
([(1,6),(1,7),(2,3),(2,5),(3,4),(4,6),(5,7)],8)=>13
([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)=>17
([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)=>10
([(0,7),(1,6),(2,5),(3,4)],8)=>11
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)=>11
([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)=>11
([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)=>21
([(0,7),(1,5),(2,4),(3,6),(4,5),(6,7)],8)=>11
([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6)],8)=>9
([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6)],8)=>9
([(0,1),(2,7),(3,7),(4,7),(5,7),(6,7)],8)=>11
([(0,3),(1,2),(4,7),(5,6),(6,7)],8)=>11
([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)=>11
([(0,1),(2,7),(3,7),(4,6),(5,6)],8)=>10
([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)=>15
([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)=>14
([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)=>11
([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(6,7),(7,8)],9)=>13
([(0,8),(1,8),(2,8),(3,8),(4,5),(5,7),(6,7),(6,8)],9)=>11
([(0,8),(1,8),(2,8),(3,4),(4,6),(5,7),(5,8),(6,7)],9)=>11
([(0,7),(1,8),(2,8),(3,4),(3,5),(4,6),(5,7),(6,8)],9)=>12
([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)=>12
([(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)=>18
([(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)=>18
([(6,7),(6,8),(7,8)],9)=>9
([(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)=>18
([(5,8),(6,7),(6,8),(7,8)],9)=>9
([(4,8),(5,8),(6,7),(6,8),(7,8)],9)=>9
([(7,8)],9)=>9
([(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)=>22
([(5,7),(5,8),(6,7),(6,8),(7,8)],9)=>10
([(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)=>10
([(6,8),(7,8)],9)=>9
([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)=>14
([(5,8),(6,8),(7,8)],9)=>9
([(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)=>18
([(4,8),(5,8),(6,8),(7,8)],9)=>9
([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)=>21
([(3,8),(4,8),(5,8),(6,8),(7,8)],9)=>10
([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)=>26
([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)=>11
([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)=>14
([],9)=>9
([(1,7),(2,8),(3,8),(4,5),(4,6),(5,7),(6,8)],9)=>10
([(2,8),(3,7),(4,5),(4,6),(5,7),(6,8)],9)=>9
([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)=>21
([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)=>26
([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7)],9)=>11
([(3,4),(5,8),(6,7),(7,8)],9)=>9
([(5,8),(6,7),(7,8)],9)=>9
([(4,8),(5,8),(6,7),(7,8)],9)=>9
([(3,8),(4,8),(5,8),(6,7),(7,8)],9)=>9
([(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)=>10
([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)=>11
([(1,8),(2,7),(3,6),(4,5)],9)=>11
([(1,4),(2,3),(5,8),(6,7),(7,8)],9)=>11
([(1,2),(3,6),(4,5),(5,7),(6,8),(7,8)],9)=>11
([(1,8),(2,7),(3,4),(3,5),(4,6),(5,7),(6,8)],9)=>11
([(0,8),(1,7),(2,3),(4,5),(4,6),(5,7),(6,8)],9)=>12
([(0,3),(1,2),(4,7),(5,6),(6,8),(7,8)],9)=>12
([(0,5),(1,4),(2,3),(6,8),(7,8)],9)=>12
([(0,6),(1,5),(2,8),(3,7),(4,7),(4,8),(5,6)],9)=>12
([(0,1),(2,8),(3,7),(4,6),(5,6),(7,8)],9)=>12
([(0,1),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)=>13
([(5,7),(5,8),(6,7),(6,8)],9)=>9
([(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)=>18
([(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)=>9
([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,4),(5,6),(7,8)],9)=>15
([(4,7),(5,6),(5,8),(6,8),(7,8)],9)=>9
([(3,7),(3,8),(4,5),(4,6),(5,6),(7,8)],9)=>10
([(4,5),(6,7),(6,8),(7,8)],9)=>9
([(3,8),(4,7),(5,6)],9)=>9
([(5,8),(6,7)],9)=>9
([(4,8),(5,7),(6,7),(6,8)],9)=>9
([(1,2),(3,8),(4,8),(5,8),(6,8),(7,8)],9)=>11
([(3,4),(5,8),(6,8),(7,8)],9)=>9
([(4,8),(5,8),(6,7)],9)=>9
([(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8)],9)=>10
([(2,3),(4,8),(5,8),(6,8),(7,8)],9)=>9
([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7)],9)=>10
([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)=>18
([(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)=>14
([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)=>10
([(3,8),(4,8),(5,7),(6,7)],9)=>9
([(2,5),(3,4),(6,8),(7,8)],9)=>9
([(2,3),(4,7),(5,6),(6,8),(7,8)],9)=>9
([(1,8),(2,8),(3,8),(4,7),(5,7),(6,7)],9)=>9
([(1,8),(2,6),(3,5),(4,7),(5,6),(7,8)],9)=>11
([(4,8),(5,6),(5,7),(6,7),(6,8),(7,8)],9)=>10
([(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)=>15
([(3,7),(3,8),(4,5),(4,6),(5,8),(6,7)],9)=>9
([(4,8),(5,7),(5,8),(6,7),(6,8)],9)=>9
([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8)],9)=>9
([(1,8),(2,8),(3,8),(4,8),(5,6),(6,7),(7,8)],9)=>11
([(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)=>18
([(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)=>18
([(7,8),(7,9),(8,9)],10)=>10
([(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)=>18
([(6,9),(7,8),(7,9),(8,9)],10)=>10
([(5,9),(6,9),(7,8),(7,9),(8,9)],10)=>10
([(8,9)],10)=>10
([(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)=>22
([(6,8),(6,9),(7,8),(7,9),(8,9)],10)=>10
([(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)=>10
([(7,9),(8,9)],10)=>10
([(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)=>14
([(6,9),(7,9),(8,9)],10)=>10
([(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)=>18
([(5,9),(6,9),(7,9),(8,9)],10)=>10
([(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)=>21
([(4,9),(5,9),(6,9),(7,9),(8,9)],10)=>10
([(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)=>26
([(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)=>11
([(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)=>14
([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)=>15
([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)=>18
([],10)=>10
([(0,8),(1,9),(2,9),(3,4),(3,5),(4,6),(5,7),(6,8),(7,9)],10)=>13
([(1,9),(2,8),(3,4),(3,5),(4,6),(5,7),(6,8),(7,9)],10)=>12
([(0,9),(1,8),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8),(7,9)],10)=>14
([(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)=>26
([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(8,9)],10)=>15
([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)=>30
([(0,9),(1,8),(2,7),(3,6),(4,5)],10)=>14
([(0,5),(1,4),(2,3),(6,9),(7,8),(8,9)],10)=>14
([(0,3),(1,2),(4,7),(5,6),(6,8),(7,9),(8,9)],10)=>14
([(0,9),(1,8),(2,3),(4,5),(4,6),(5,7),(6,8),(7,9)],10)=>14
([(1,9),(2,8),(3,4),(5,6),(5,7),(6,8),(7,9)],10)=>12
([(1,4),(2,3),(5,8),(6,7),(7,9),(8,9)],10)=>12
([(1,6),(2,5),(3,4),(7,9),(8,9)],10)=>12
([(2,9),(3,8),(4,7),(5,6)],10)=>11
([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)=>14
([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(8,9)],10)=>15
([(6,8),(6,9),(7,8),(7,9)],10)=>10
([(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9)],10)=>18
([(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)=>10
([(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)=>21
([(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)=>14
([(5,8),(6,7),(6,9),(7,9),(8,9)],10)=>10
([(6,9),(7,8),(8,9)],10)=>10
([(1,8),(1,9),(2,6),(2,7),(3,4),(3,5),(4,5),(6,7),(8,9)],10)=>15
([(4,8),(4,9),(5,6),(5,7),(6,7),(8,9)],10)=>10
([(5,6),(7,8),(7,9),(8,9)],10)=>10
([(4,9),(5,8),(6,7)],10)=>10
([(6,9),(7,8)],10)=>10
([(5,9),(6,8),(7,8),(7,9)],10)=>10
([(0,1),(2,9),(3,7),(4,6),(5,8),(6,7),(8,9)],10)=>14
([(0,7),(1,6),(2,9),(3,8),(4,5),(4,8),(5,9),(6,7)],10)=>14
([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(8,9)],10)=>14
([(2,3),(4,9),(5,9),(6,8),(7,8)],10)=>10
([(3,4),(5,9),(6,9),(7,9),(8,9)],10)=>10
([(3,9),(4,9),(5,9),(6,8),(7,8)],10)=>10
([(0,8),(1,9),(2,9),(3,9),(4,5),(4,6),(5,7),(6,8),(7,9)],10)=>13
([(0,9),(1,9),(2,9),(3,9),(4,5),(5,7),(6,8),(6,9),(7,8)],10)=>13
([(0,9),(1,9),(2,9),(3,9),(4,9),(5,6),(6,8),(7,8),(7,9)],10)=>13
([(0,1),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)=>15
([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8)],10)=>13
([(0,9),(1,9),(2,9),(3,9),(4,9),(5,8),(6,8),(7,8)],10)=>11
([(5,9),(6,9),(7,8)],10)=>10
([(2,5),(3,4),(6,9),(7,8),(8,9)],10)=>11
([(3,9),(4,8),(5,6),(5,7),(6,8),(7,9)],10)=>10
([(4,9),(5,8),(6,7),(8,9)],10)=>10
([(4,8),(4,9),(5,6),(5,7),(6,9),(7,8)],10)=>10
([(5,9),(6,8),(6,9),(7,8),(7,9)],10)=>10
([(1,2),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)=>13
([(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(8,9)],10)=>11
([(5,9),(6,9),(7,8),(8,9)],10)=>10
([(0,10),(1,9),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10)],11)=>15
([(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)=>18
([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)=>19
([(9,10)],11)=>11
([(0,9),(1,10),(2,10),(3,4),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10)],11)=>15
([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(9,10)],11)=>18
([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)=>15
([],11)=>11
([(8,10),(9,10)],11)=>11
([(0,9),(1,10),(2,10),(3,10),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10)],11)=>14
([(0,10),(1,10),(2,10),(3,10),(4,5),(5,9),(6,8),(6,10),(7,8),(7,9)],11)=>14
([(0,10),(1,10),(2,10),(3,10),(4,10),(5,6),(6,8),(7,9),(7,10),(8,9)],11)=>15
([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,7),(7,9),(8,9),(8,10)],11)=>15
([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,8),(8,9),(9,10)],11)=>17
([(0,9),(0,10),(1,9),(1,10),(2,9),(2,10),(3,9),(3,10),(4,9),(4,10),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)=>33
([(1,10),(2,9),(3,8),(4,7),(5,6)],11)=>14
([(1,6),(2,5),(3,4),(7,10),(8,9),(9,10)],11)=>14
([(1,9),(1,10),(2,9),(2,10),(3,9),(3,10),(4,9),(4,10),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)=>30
([(5,10),(6,9),(7,8),(9,10)],11)=>11
([(0,1),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)=>17
([(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(9,10)],11)=>15
([(6,10),(7,9),(7,10),(8,9),(8,10)],11)=>11
([(7,10),(8,9),(8,10),(9,10)],11)=>11
([(8,9),(8,10),(9,10)],11)=>11
([(1,4),(2,3),(5,8),(6,7),(7,9),(8,10),(9,10)],11)=>14
([(6,10),(7,10),(8,9),(8,10),(9,10)],11)=>11
([(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(9,10)],11)=>14
([(5,10),(6,10),(7,10),(8,9),(9,10)],11)=>11
([(0,11),(1,10),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>17
([(10,11)],12)=>12
([(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>12
([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,10),(10,11)],12)=>19
([(2,11),(3,10),(4,9),(5,8),(6,7)],12)=>14
([(0,11),(1,10),(2,9),(3,8),(4,7),(5,6)],12)=>17
([(0,10),(1,11),(2,11),(3,4),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>16
([],12)=>12
([(8,11),(9,10),(9,11),(10,11)],12)=>12
([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>22
([(0,10),(0,11),(1,10),(1,11),(2,10),(2,11),(3,10),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)=>38
([(2,10),(2,11),(3,10),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)=>30
([(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)=>21
([(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)=>14
([(8,10),(8,11),(9,10),(9,11)],12)=>12
([(0,7),(1,6),(2,5),(3,4),(8,11),(9,10),(10,11)],12)=>17
([(0,5),(1,4),(2,3),(6,9),(7,8),(8,10),(9,11),(10,11)],12)=>17
([(0,3),(1,2),(4,11),(5,9),(6,8),(7,10),(8,9),(10,11)],12)=>17
([(0,3),(1,2),(4,7),(5,6),(6,10),(7,11),(8,9),(8,10),(9,11)],12)=>17
([(0,1),(2,7),(3,6),(4,9),(5,8),(6,7),(8,10),(9,11),(10,11)],12)=>17
([(0,11),(1,10),(2,3),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>17
([(0,11),(1,9),(2,7),(3,6),(4,8),(5,10),(6,7),(8,9),(10,11)],12)=>17
([(0,9),(1,8),(2,11),(3,10),(4,5),(4,6),(5,7),(6,8),(7,9),(10,11)],12)=>17
([(0,11),(1,9),(2,8),(3,10),(4,5),(4,8),(5,9),(6,7),(6,10),(7,11)],12)=>17
([(1,8),(2,7),(3,6),(4,5),(9,11),(10,11)],12)=>15
([(2,5),(3,4),(6,11),(7,11),(8,10),(9,10)],12)=>13
([(4,5),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>12
([(3,11),(4,11),(5,10),(6,10),(7,9),(8,9)],12)=>12
([(4,11),(5,11),(6,11),(7,11),(8,10),(9,10)],12)=>12
([(4,11),(5,11),(6,11),(7,10),(8,10),(9,10)],12)=>12
([(9,11),(10,11)],12)=>12
([(0,10),(0,11),(1,7),(1,8),(2,4),(2,5),(3,4),(3,5),(6,7),(6,8),(9,10),(9,11)],12)=>17
([(0,1),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>19
([(0,11),(1,11),(2,11),(3,10),(4,10),(5,10),(6,9),(7,9),(8,9)],12)=>14
([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,10),(8,10),(9,10)],12)=>15
([(1,11),(2,10),(3,4),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>15
([(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>19
([(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>14
([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,10),(9,10),(10,11)],12)=>18
([(4,11),(5,10),(6,9),(7,8)],12)=>12
([(0,11),(0,12),(1,11),(1,12),(2,11),(2,12),(3,11),(3,12),(4,11),(4,12),(5,11),(5,12),(6,11),(6,12),(7,11),(7,12),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12)],13)=>42
([(0,10),(1,8),(2,7),(3,9),(4,12),(5,11),(6,11),(6,12),(7,8),(9,10)],13)=>18
([(5,10),(6,10),(7,10),(8,9)],11)=>11
([(4,10),(5,9),(6,7),(6,8),(7,9),(8,10)],11)=>11
([(6,11),(7,10),(8,9)],12)=>12
([(2,10),(3,10),(4,10),(5,10),(6,7),(7,9),(8,9),(8,10)],11)=>11
([(7,11),(8,11),(9,10),(9,11),(10,11)],12)=>12
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,12),(8,12),(9,11),(10,11),(11,12)],13)=>19
([(1,2),(3,10),(4,8),(5,7),(6,9),(7,8),(9,10)],11)=>14
([(4,9),(4,10),(5,6),(5,8),(6,7),(7,9),(8,10)],11)=>13
([(5,10),(6,10),(7,10),(8,10),(9,10)],11)=>11
([(3,11),(4,11),(5,10),(6,9),(7,8),(7,9),(8,10)],12)=>12
([(4,9),(5,10),(6,10),(7,10),(8,9),(8,10)],11)=>11
([(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)=>10
([(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)=>14
([(1,8),(1,9),(2,3),(2,5),(3,4),(4,6),(5,7),(6,8),(7,9)],10)=>17
([(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)=>11
([(0,12),(1,11),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11),(10,12)],13)=>18
([(0,10),(1,10),(2,10),(3,10),(4,10),(5,9),(6,9),(7,9),(8,9)],11)=>13
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)=>23
([(0,10),(1,10),(2,9),(3,9),(4,8),(5,8),(6,12),(7,11),(11,12)],13)=>17
([(3,9),(4,9),(5,9),(6,7),(7,8),(8,9)],10)=>10
([(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,10),(10,11)],12)=>15
([(1,10),(2,9),(3,4),(5,6),(5,7),(6,8),(7,9),(8,10)],11)=>14
([(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)=>14
([(4,10),(5,11),(6,11),(7,11),(8,9),(8,10),(9,11)],12)=>12
([(5,10),(6,10),(7,9),(8,9)],11)=>11
([(1,7),(1,8),(2,3),(2,4),(3,6),(4,5),(5,7),(6,8)],9)=>11
([(5,10),(6,10),(7,9),(8,9),(9,10)],11)=>11
([(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)=>10
([(3,7),(4,8),(5,8),(6,7),(6,8)],9)=>9
([(0,12),(1,12),(2,11),(3,10),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11)],13)=>18
([(1,2),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9)],12)=>14
([(1,9),(2,9),(3,9),(4,9),(5,6),(6,8),(7,8),(7,9)],10)=>11
([(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)=>18
([(2,7),(3,6),(4,5),(8,10),(9,10)],11)=>12
([(4,5),(6,10),(7,10),(8,10),(9,10)],11)=>11
([(1,9),(2,8),(3,11),(4,10),(5,7),(5,10),(6,7),(6,11),(8,9)],12)=>15
([(0,10),(1,9),(2,12),(3,11),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10),(11,12)],13)=>18
([(6,9),(6,10),(7,8),(7,10),(8,9),(9,10)],11)=>11
([(7,9),(7,10),(8,9),(8,10)],11)=>11
([(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,10),(9,10)],12)=>15
([(1,10),(2,10),(3,10),(4,10),(5,10),(6,9),(7,9),(8,9)],11)=>11
([(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,8),(8,10),(9,10),(9,11)],12)=>15
([(0,3),(1,2),(4,10),(5,9),(6,8),(7,8),(9,11),(10,12),(11,12)],13)=>18
([(0,1),(2,7),(3,6),(4,9),(5,8),(6,7),(8,10),(9,10)],11)=>15
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,8),(8,10),(9,11),(9,12),(10,11)],13)=>19
([(7,8),(9,10),(9,11),(10,11)],12)=>12
([(5,9),(5,10),(6,7),(6,8),(7,8),(9,10)],11)=>11
([(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)=>18
([(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)=>11
([(5,11),(6,11),(7,11),(8,11),(9,10),(10,11)],12)=>12
([(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)=>18
([(6,10),(7,10),(8,9)],11)=>11
([(0,12),(1,11),(2,5),(3,4),(6,7),(6,8),(7,9),(8,10),(9,11),(10,12)],13)=>18
([(4,10),(5,9),(6,7),(8,9),(8,10)],11)=>11
([(0,10),(1,9),(2,12),(3,11),(4,5),(6,8),(6,11),(7,8),(7,12),(9,10)],13)=>18
([(2,11),(3,10),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>14
([(1,11),(2,11),(3,10),(4,9),(5,6),(5,7),(6,8),(7,9),(8,10)],12)=>15
([(7,11),(8,11),(9,11),(10,11)],12)=>12
([(0,11),(0,12),(1,2),(1,4),(2,3),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11),(10,12)],13)=>25
([(2,9),(2,10),(3,7),(3,8),(4,5),(4,6),(5,6),(7,8),(9,10)],11)=>15
([(0,9),(0,10),(1,2),(1,4),(2,3),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10)],11)=>21
([(1,10),(2,11),(3,11),(4,11),(5,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>14
([(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)=>18
([(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,10),(9,10)],12)=>13
([(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,10),(9,10)],11)=>15
([(5,9),(6,8),(7,8),(7,9),(8,9)],10)=>10
([(0,10),(1,9),(2,3),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10)],11)=>15
([(0,10),(1,9),(2,12),(3,11),(4,7),(4,8),(5,6),(5,9),(6,10),(7,11),(8,12)],13)=>18
([(3,11),(4,11),(5,11),(6,7),(7,9),(8,10),(8,11),(9,10)],12)=>12
([(4,7),(4,8),(5,6),(5,8),(6,7)],9)=>9
([(2,9),(3,8),(4,11),(5,10),(6,10),(6,11),(7,8),(7,9)],12)=>13
([(2,8),(2,9),(3,5),(3,6),(4,5),(4,6),(7,8),(7,9)],10)=>11
([(6,11),(7,11),(8,11),(9,10),(10,11)],12)=>12
([(3,10),(4,10),(5,10),(6,10),(7,9),(8,9)],11)=>11
([(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,10),(10,11)],12)=>14
([(3,11),(4,11),(5,11),(6,11),(7,11),(8,9),(9,10),(10,11)],12)=>13
([(0,13),(0,14),(1,13),(1,14),(2,13),(2,14),(3,13),(3,14),(4,13),(4,14),(5,13),(5,14),(6,13),(6,14),(7,13),(7,14),(8,13),(8,14),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14)],15)=>50
([(3,11),(4,11),(5,11),(6,11),(7,11),(8,10),(9,10)],12)=>12
([(8,11),(9,10)],12)=>12
([(3,10),(4,10),(5,10),(6,9),(7,9),(8,9)],11)=>11
([(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10)],11)=>11
([(4,9),(4,10),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)=>21
([(0,1),(2,12),(3,12),(4,12),(5,12),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)=>21
([(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)=>10
([(1,9),(2,9),(3,9),(4,9),(5,8),(6,8),(7,8)],10)=>10
([(4,11),(5,10),(6,7),(6,8),(7,9),(8,10),(9,11)],12)=>12
([(2,8),(3,9),(4,9),(5,6),(5,7),(6,8),(7,9)],10)=>10
([(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)=>9
([(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9)],11)=>13
([(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)=>14
([(3,9),(3,10),(4,9),(4,10),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)=>26
([(0,8),(1,7),(2,10),(3,9),(4,6),(5,6),(7,8),(9,11),(10,11)],12)=>16
([(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,11),(10,11)],12)=>17
([(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)=>15
([(5,11),(6,10),(7,8),(9,10),(9,11)],12)=>12
([(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)=>18
([(2,10),(3,11),(4,11),(5,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>13
([(0,10),(1,10),(2,9),(3,8),(4,5),(4,6),(5,7),(6,8),(7,9)],11)=>15
([(1,8),(2,8),(3,7),(4,7),(5,10),(6,9),(9,11),(10,11)],12)=>14
([(0,11),(1,11),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8)],12)=>15
([(0,11),(1,12),(2,12),(3,12),(4,12),(5,6),(5,7),(6,8),(7,9),(8,10),(9,11),(10,12)],13)=>17
([(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)=>18
([(0,1),(2,8),(3,7),(4,6),(5,6),(7,10),(8,11),(9,10),(9,11)],12)=>16
([(2,10),(3,9),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10)],11)=>12
([(1,11),(2,11),(3,11),(4,11),(5,11),(6,7),(7,9),(8,10),(8,11),(9,10)],12)=>15
([(2,7),(3,6),(4,5),(8,11),(9,10),(10,11)],12)=>14
([(0,9),(1,8),(2,11),(3,10),(4,12),(5,12),(6,7),(6,10),(7,11),(8,9)],13)=>18
([(7,10),(8,9)],11)=>11
([(4,7),(4,8),(5,6),(5,8),(6,8),(7,8)],9)=>9
([(1,8),(2,8),(3,8),(4,8),(5,7),(6,7)],9)=>9
([(0,5),(1,4),(2,3),(6,9),(7,8),(8,11),(9,12),(10,11),(10,12)],13)=>18
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,12),(8,12),(9,11),(10,11)],13)=>19
([(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9)],11)=>11
([(8,11),(9,10),(10,11)],12)=>12
([(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,10),(10,11)],12)=>18
([(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)=>12
([(2,11),(3,11),(4,11),(5,11),(6,7),(7,9),(8,10),(8,11),(9,10)],12)=>13
([(0,3),(1,2),(4,10),(5,9),(6,8),(7,8),(9,10)],11)=>15
([(0,7),(1,6),(2,5),(3,4),(8,10),(9,10)],11)=>15
([(3,9),(4,10),(5,10),(6,7),(6,8),(7,9),(8,10)],11)=>11
([(5,8),(6,7),(9,11),(10,11)],12)=>12
([(0,11),(1,12),(2,12),(3,12),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11),(10,12)],13)=>17
([(2,8),(3,8),(4,8),(5,7),(6,7)],9)=>9
([(1,11),(2,9),(3,8),(4,10),(5,7),(6,7),(8,9),(10,11)],12)=>15
([(2,9),(3,8),(4,11),(5,10),(6,7),(6,10),(7,11),(8,9)],12)=>14
([(1,10),(2,10),(3,10),(4,10),(5,10),(6,7),(7,9),(8,9),(8,10)],11)=>13
([(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)=>18
([(6,11),(7,11),(8,10),(8,11),(9,10),(9,11)],12)=>12
([(5,11),(6,11),(7,11),(8,10),(9,10)],12)=>12
([(4,8),(5,7),(6,7),(6,8),(7,8)],9)=>9
([(1,4),(2,3),(5,8),(6,7),(7,10),(8,11),(9,10),(9,11)],12)=>15
([(2,9),(3,10),(4,10),(5,6),(5,7),(6,8),(7,9),(8,10)],11)=>12
([(5,9),(6,7),(6,8),(7,8),(7,9),(8,9)],10)=>10
([(6,11),(7,11),(8,10),(9,10),(10,11)],12)=>12
([(5,11),(6,11),(7,11),(8,10),(9,10),(10,11)],12)=>12
([(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)=>11
([(2,10),(2,11),(3,4),(3,5),(4,7),(5,6),(6,8),(7,9),(8,10),(9,11)],12)=>14
([(1,9),(2,9),(3,9),(4,5),(5,7),(6,8),(6,9),(7,8)],10)=>11
([(3,11),(4,11),(5,11),(6,11),(7,11),(8,10),(9,10),(10,11)],12)=>12
([(0,1),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8)],11)=>14
([(5,10),(6,11),(7,11),(8,11),(9,10),(9,11)],12)=>12
([(6,10),(7,9),(8,9),(8,10),(9,10)],11)=>11
([(0,8),(1,7),(2,10),(3,9),(4,7),(4,8),(5,6),(5,9),(6,10)],11)=>15
([(6,11),(7,10),(8,9),(10,11)],12)=>12
([(6,10),(7,11),(8,11),(9,10),(9,11)],12)=>12
([(2,10),(3,10),(4,10),(5,10),(6,10),(7,8),(8,9),(9,10)],11)=>13
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,12),(8,12),(9,12),(10,11),(11,12)],13)=>22
([(4,10),(5,10),(6,9),(7,8),(8,9)],11)=>11
([(2,10),(3,10),(4,10),(5,6),(6,8),(7,9),(7,10),(8,9)],11)=>11
([(1,4),(2,3),(5,10),(6,10),(7,9),(8,9)],11)=>13
([(2,9),(3,9),(4,9),(5,6),(6,8),(7,8),(7,9)],10)=>10
([(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10)],11)=>18
([(4,9),(4,10),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)=>18
([(3,10),(4,9),(5,8),(6,7)],11)=>11
([(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>18
([(0,3),(1,2),(4,10),(5,9),(6,8),(7,8),(9,11),(10,11)],12)=>16
([(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11)],12)=>18
([(0,9),(1,8),(2,11),(3,10),(4,12),(5,12),(6,10),(6,11),(7,8),(7,9)],13)=>17
([(1,2),(3,9),(4,8),(5,7),(6,7),(8,10),(9,10)],11)=>13
([(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)=>18
([(4,10),(4,11),(5,7),(5,8),(6,7),(6,8),(9,10),(9,11)],12)=>12
([(0,10),(1,11),(2,11),(3,11),(4,11),(5,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>15
([(0,10),(1,8),(2,7),(3,9),(4,6),(5,6),(7,8),(9,10)],11)=>15
([(3,10),(4,11),(5,11),(6,7),(6,8),(7,9),(8,10),(9,11)],12)=>12
([(6,11),(7,11),(8,11),(9,10)],12)=>12
([(1,2),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>17
([(6,9),(6,10),(7,8),(7,10),(8,9)],11)=>11
([(0,10),(0,11),(1,8),(1,9),(2,6),(2,7),(3,4),(3,5),(4,5),(6,7),(8,9),(10,11)],12)=>20
([(4,5),(6,11),(7,11),(8,10),(9,10)],12)=>12
([(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)=>22
([(0,7),(0,8),(1,2),(1,4),(2,3),(3,5),(4,6),(5,7),(6,8)],9)=>17
([(3,9),(4,8),(5,7),(6,7),(8,10),(9,10)],11)=>11
([(3,10),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)=>26
([(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)=>18
([(2,9),(2,10),(3,9),(3,10),(4,9),(4,10),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)=>26
([(3,10),(4,10),(5,10),(6,7),(7,9),(8,9),(8,10)],11)=>11
([(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(6,7),(6,8)],11)=>13
([(2,8),(3,8),(4,8),(5,6),(6,7),(7,8)],9)=>9
([(7,10),(7,11),(8,9),(8,11),(9,10),(10,11)],12)=>12
([(2,8),(3,7),(4,10),(5,9),(6,9),(6,10),(7,8)],11)=>12
([(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)=>15
([(1,8),(2,8),(3,8),(4,5),(5,7),(6,7),(6,8)],9)=>10
([(4,10),(5,10),(6,10),(7,9),(8,9)],11)=>11
([(6,10),(6,11),(7,8),(7,9),(8,9),(10,11)],12)=>12
([(1,8),(2,7),(3,10),(4,9),(5,6),(5,9),(6,10),(7,8)],11)=>14
([(3,11),(4,10),(5,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>12
([(7,10),(8,9),(8,11),(9,11),(10,11)],12)=>12
([(2,11),(3,10),(4,5),(6,7),(6,8),(7,9),(8,10),(9,11)],12)=>14
([(2,10),(3,10),(4,9),(5,9),(6,8),(7,8)],11)=>11
([(0,12),(1,12),(2,11),(3,10),(4,5),(6,7),(6,8),(7,9),(8,10),(9,11)],13)=>18
([(1,9),(1,10),(2,3),(2,4),(3,6),(4,5),(5,7),(6,8),(7,9),(8,10)],11)=>14
([(1,8),(2,8),(3,7),(4,7),(5,10),(6,9),(9,10)],11)=>13
([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,9),(9,10),(10,11)],12)=>19
([(0,9),(1,8),(2,11),(3,10),(4,5),(4,6),(5,8),(6,9),(7,10),(7,11)],12)=>16
([(2,9),(3,9),(4,9),(5,9),(6,8),(7,8)],10)=>10
([(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8)],10)=>11
([(0,8),(1,7),(2,10),(3,9),(4,6),(4,9),(5,6),(5,10),(7,8)],11)=>15
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,11),(8,11),(9,11),(10,11)],13)=>15
([(0,15),(0,16),(1,15),(1,16),(2,15),(2,16),(3,15),(3,16),(4,15),(4,16),(5,15),(5,16),(6,15),(6,16),(7,15),(7,16),(8,15),(8,16),(9,15),(9,16),(10,15),(10,16),(11,15),(11,16),(12,15),(12,16),(13,15),(13,16),(14,15),(14,16)],17)=>57
([(5,10),(6,11),(7,11),(8,9),(8,10),(9,11)],12)=>12
([(0,8),(0,9),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7),(6,8),(7,9)],10)=>14
([(0,1),(2,12),(3,10),(4,9),(5,11),(6,8),(7,8),(9,10),(11,12)],13)=>18
([(3,11),(4,11),(5,11),(6,11),(7,10),(8,10),(9,10)],12)=>12
([(2,3),(4,10),(5,9),(6,8),(7,8),(9,11),(10,11)],12)=>13
([(2,10),(3,9),(4,5),(6,7),(6,8),(7,9),(8,10)],11)=>12
([(0,14),(0,15),(1,14),(1,15),(2,14),(2,15),(3,14),(3,15),(4,14),(4,15),(5,14),(5,15),(6,14),(6,15),(7,14),(7,15),(8,14),(8,15),(9,14),(9,15),(10,14),(10,15),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15)],16)=>54
([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,10),(9,10)],12)=>17
([(4,9),(5,10),(6,10),(7,8),(7,9),(8,10)],11)=>11
([(0,12),(1,12),(2,11),(3,11),(4,10),(5,9),(6,7),(6,8),(7,9),(8,10)],13)=>17
([(4,11),(5,11),(6,11),(7,11),(8,11),(9,10),(10,11)],12)=>12
([(1,6),(2,5),(3,4),(7,10),(8,9),(9,11),(10,11)],12)=>15
([(0,5),(1,4),(2,3),(6,12),(7,11),(8,10),(9,10),(11,12)],13)=>18
([(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>12
([(4,11),(5,9),(6,8),(7,10),(8,9),(10,11)],12)=>12
([(4,5),(6,9),(7,9),(8,9)],10)=>10
([(5,10),(5,11),(6,7),(6,9),(7,8),(8,10),(9,11)],12)=>13
([(2,8),(3,8),(4,5),(5,7),(6,7),(6,8)],9)=>9
([(1,11),(2,11),(3,11),(4,11),(5,6),(6,10),(7,9),(7,11),(8,9),(8,10)],12)=>14
([(3,6),(4,5),(7,10),(8,9),(9,11),(10,11)],12)=>12
([(3,10),(4,10),(5,10),(6,10),(7,8),(8,9),(9,10)],11)=>11
([(2,11),(3,11),(4,10),(5,9),(6,7),(6,8),(7,9),(8,10)],12)=>13
([(6,9),(6,10),(7,8),(7,10),(8,9),(8,10),(9,10)],11)=>11
([(2,8),(2,9),(3,4),(3,5),(4,7),(5,6),(6,8),(7,9)],10)=>11
([(7,10),(8,9),(9,10)],11)=>11
([(4,9),(5,9),(6,8),(7,8),(8,9)],10)=>10
([(7,11),(8,11),(9,10),(10,11)],12)=>12
([(6,9),(7,8),(7,10),(8,10),(9,10)],11)=>11
([(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)=>17
([(4,10),(5,10),(6,10),(7,10),(8,9),(9,10)],11)=>11
([(6,7),(8,9),(8,10),(9,10)],11)=>11
([(2,9),(3,9),(4,9),(5,8),(6,8),(7,8)],10)=>10
([(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)=>12
([(6,10),(6,11),(7,8),(7,9),(8,11),(9,10)],12)=>12
([(7,10),(8,10),(9,10)],11)=>11
([(4,11),(5,11),(6,11),(7,11),(8,9),(9,10),(10,11)],12)=>12
([(5,11),(6,11),(7,10),(8,9),(9,10)],12)=>12
([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,10),(7,10),(8,10),(9,10)],12)=>14
([(3,8),(3,9),(4,5),(4,7),(5,6),(6,8),(7,9)],10)=>13
([(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,11),(10,11)],12)=>15
([(1,2),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)=>15
([(5,9),(5,10),(6,7),(6,8),(7,10),(8,9)],11)=>11
([(3,10),(4,8),(5,7),(6,9),(7,8),(9,10)],11)=>11
([(4,7),(5,6),(8,10),(9,10)],11)=>11
([(0,1),(2,7),(3,6),(4,9),(5,8),(6,11),(7,11),(8,10),(9,10)],12)=>16
([(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,9),(9,10),(10,11)],12)=>15
([(5,8),(5,9),(6,7),(6,9),(7,8)],10)=>10
([(1,9),(2,8),(3,11),(4,10),(5,8),(5,9),(6,7),(6,10),(7,11)],12)=>15
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,12),(8,9),(9,11),(10,11),(10,12)],13)=>19
([(7,11),(8,10),(9,10),(9,11),(10,11)],12)=>12
([(6,11),(7,11),(8,10),(9,10)],12)=>12
([(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,8),(7,8)],9)=>17
([(1,7),(1,8),(2,4),(2,5),(3,4),(3,5),(6,7),(6,8)],9)=>11
([(3,9),(4,9),(5,9),(6,9),(7,8),(8,9)],10)=>10
([(3,4),(5,11),(6,10),(7,9),(8,9),(10,11)],12)=>12
([(1,11),(2,11),(3,11),(4,11),(5,11),(6,10),(7,10),(8,10),(9,10)],12)=>13
([(9,10),(9,11),(10,11)],12)=>12
([(3,8),(4,8),(5,7),(6,7),(7,8)],9)=>9
([(1,10),(1,11),(2,10),(2,11),(3,10),(3,11),(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)=>33
([(0,5),(1,4),(2,3),(6,11),(7,11),(8,10),(9,10)],12)=>16
([(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9)],10)=>10
([(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)=>10
([(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,9),(9,10),(10,11)],12)=>17
([(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)=>11
([(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)=>14
([(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)=>14
([(0,1),(2,9),(3,9),(4,8),(5,8),(6,11),(7,10),(10,11)],12)=>16
([(4,11),(5,10),(6,7),(8,9),(8,10),(9,11)],12)=>12
([(0,11),(1,12),(2,12),(3,4),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11),(10,12)],13)=>18
([(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,10),(9,10),(10,11)],12)=>14
([(1,10),(1,11),(2,3),(2,5),(3,4),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>21
([(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)=>11
([(0,5),(1,4),(2,3),(6,9),(7,8),(8,10),(9,10)],11)=>15
([(0,16),(0,17),(1,16),(1,17),(2,16),(2,17),(3,16),(3,17),(4,16),(4,17),(5,16),(5,17),(6,16),(6,17),(7,16),(7,17),(8,16),(8,17),(9,16),(9,17),(10,16),(10,17),(11,16),(11,17),(12,16),(12,17),(13,16),(13,17),(14,16),(14,17),(15,16),(15,17)],18)=>62
([(5,9),(6,10),(7,10),(8,9),(8,10)],11)=>11
([(4,8),(5,9),(6,9),(7,8),(7,9)],10)=>10
([(0,10),(0,11),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>17
([(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>15
([(1,2),(3,8),(4,7),(5,10),(6,9),(7,8),(9,11),(10,11)],12)=>15
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,6),(6,11),(7,10),(7,12),(8,9),(8,10),(9,11)],13)=>17
([(2,11),(3,11),(4,11),(5,11),(6,11),(7,8),(8,10),(9,10),(9,11)],12)=>13
([(1,10),(2,11),(3,11),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>15
([(7,10),(7,11),(8,9),(8,11),(9,10)],12)=>12
([(0,11),(1,11),(2,11),(3,11),(4,11),(5,6),(6,10),(7,9),(7,11),(8,9),(8,10)],12)=>15
([(3,4),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>13
([(2,5),(3,4),(6,9),(7,8),(8,10),(9,10)],11)=>12
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,11),(7,11),(8,11),(9,11),(10,11)],13)=>16
([(0,7),(1,7),(2,6),(3,6),(4,9),(5,8),(8,10),(9,10)],11)=>14
([(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)=>11
([(3,11),(4,10),(5,6),(7,8),(7,9),(8,10),(9,11)],12)=>12
([(1,4),(2,3),(5,11),(6,10),(7,9),(8,9),(10,11)],12)=>15
([(3,10),(4,9),(5,6),(5,7),(6,8),(7,9),(8,10)],11)=>11
([(0,7),(1,7),(2,6),(3,6),(4,9),(5,8),(8,10),(9,11),(10,11)],12)=>16
([(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)=>12
([(2,10),(3,10),(4,9),(5,8),(6,7),(6,8),(7,9)],11)=>12
([(2,9),(3,9),(4,8),(5,8),(6,11),(7,10),(10,11)],12)=>13
([(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)=>18
([(0,3),(1,2),(4,12),(5,12),(6,11),(7,11),(8,10),(9,10)],13)=>17
([(0,10),(1,9),(2,12),(3,11),(4,5),(6,9),(6,10),(7,8),(7,11),(8,12)],13)=>18
([(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,10),(9,10),(10,11)],12)=>15
([(4,7),(5,6),(8,11),(9,10),(10,11)],12)=>12
([(0,12),(1,11),(2,3),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11),(10,12)],13)=>18
([(4,10),(4,11),(5,6),(5,7),(6,9),(7,8),(8,10),(9,11)],12)=>12
([(1,2),(3,9),(4,8),(5,7),(6,7),(8,10),(9,11),(10,11)],12)=>15
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,7),(7,11),(8,10),(8,12),(9,10),(9,11)],13)=>17
([(1,10),(2,10),(3,9),(4,8),(5,6),(5,7),(6,8),(7,9)],11)=>13
([(6,10),(7,10),(8,10),(9,10)],11)=>11
([(7,9),(7,10),(8,9),(8,10),(9,10)],11)=>11
([(0,12),(0,13),(1,12),(1,13),(2,12),(2,13),(3,12),(3,13),(4,12),(4,13),(5,12),(5,13),(6,12),(6,13),(7,12),(7,13),(8,12),(8,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13)],14)=>46
([(6,10),(7,8),(7,9),(8,9),(8,10),(9,10)],11)=>11
([(2,3),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)=>13
([(2,5),(3,4),(6,9),(7,8),(8,10),(9,11),(10,11)],12)=>14
([(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)=>12
([(3,8),(4,9),(5,9),(6,7),(6,8),(7,9)],10)=>10
([(3,8),(4,7),(5,6),(9,11),(10,11)],12)=>12
([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9)],11)=>15
([(2,11),(3,11),(4,11),(5,11),(6,11),(7,10),(8,10),(9,10)],12)=>12
([(2,3),(4,10),(5,9),(6,8),(7,8),(9,10)],11)=>12
([(3,4),(5,8),(6,7),(7,9),(8,10),(9,10)],11)=>11
([(7,11),(8,11),(9,10)],12)=>12
([(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,8),(8,9),(9,10)],11)=>15
([(3,4),(5,10),(6,10),(7,9),(8,9)],11)=>11
([(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)=>11
([(4,9),(5,9),(6,9),(7,8),(8,9)],10)=>10
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,12),(8,12),(9,10),(10,11),(11,12)],13)=>21
([(0,1),(2,9),(3,9),(4,8),(5,8),(6,11),(7,10),(10,12),(11,12)],13)=>17
([(2,3),(4,9),(5,9),(6,9),(7,9),(8,9)],10)=>11
([(5,11),(6,10),(7,9),(7,10),(8,9),(8,11)],12)=>12
([(5,10),(6,9),(7,8)],11)=>11
([(1,9),(2,10),(3,10),(4,10),(5,6),(5,7),(6,8),(7,9),(8,10)],11)=>13
([(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)=>10
([(0,3),(1,2),(4,9),(5,8),(6,11),(7,10),(8,9),(10,12),(11,12)],13)=>18
([(3,9),(3,10),(4,9),(4,10),(5,9),(5,10),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10)],11)=>21
([(4,10),(4,11),(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)=>26
([(3,9),(3,10),(4,5),(4,6),(5,8),(6,7),(7,9),(8,10)],11)=>11
([(5,10),(6,9),(7,8),(7,9),(8,10)],11)=>11
([(3,10),(3,11),(4,8),(4,9),(5,6),(5,7),(6,7),(8,9),(10,11)],12)=>15
([(1,9),(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(8,9)],10)=>13
([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,8),(8,10),(9,10),(9,11)],12)=>17
([(6,11),(7,10),(8,9),(8,10),(9,11)],12)=>12
([(3,10),(3,11),(4,5),(4,7),(5,6),(6,8),(7,9),(8,10),(9,11)],12)=>17
([(3,11),(4,11),(5,11),(6,11),(7,8),(8,10),(9,10),(9,11)],12)=>12
([(3,9),(4,8),(5,11),(6,10),(7,10),(7,11),(8,9)],12)=>12
([(5,6),(7,11),(8,11),(9,11),(10,11)],12)=>12
([(6,10),(7,10),(8,9),(9,10)],11)=>11
([(4,9),(5,9),(6,8),(6,9),(7,8),(7,9)],10)=>10
([(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(9,10)],11)=>11
([(2,9),(2,10),(3,4),(3,6),(4,5),(5,7),(6,8),(7,9),(8,10)],11)=>17
([(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,10),(8,10),(9,10)],12)=>13
([(5,10),(5,11),(6,10),(6,11),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)=>21
([(4,11),(5,11),(6,10),(7,9),(8,9),(8,10)],12)=>12
([(0,3),(1,2),(4,7),(5,6),(6,9),(7,10),(8,9),(8,10)],11)=>15
([(7,11),(8,10),(9,10),(9,11)],12)=>12
([(5,10),(6,10),(7,9),(7,10),(8,9),(8,10)],11)=>11
([(1,11),(2,10),(3,4),(5,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>15
([(1,9),(2,10),(3,10),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10)],11)=>13
([(3,6),(4,5),(7,10),(8,9),(9,10)],11)=>11
([(7,11),(8,10),(8,11),(9,10),(9,11)],12)=>12
([(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)=>17
([(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)=>22
([(0,9),(1,8),(2,7),(3,6),(4,5),(10,12),(11,12)],13)=>18
([(2,7),(2,8),(3,4),(3,6),(4,5),(5,7),(6,8)],9)=>13
([(2,3),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)=>15
([(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)=>12
([(3,4),(5,10),(6,10),(7,10),(8,10),(9,10)],11)=>11
([(0,1),(2,8),(3,7),(4,6),(5,6),(7,9),(8,10),(9,10)],11)=>15
([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,9),(7,9),(8,9)],11)=>13
([(2,10),(3,10),(4,10),(5,10),(6,9),(7,9),(8,9)],11)=>11
([(2,10),(3,11),(4,11),(5,11),(6,7),(6,8),(7,9),(8,10),(9,11)],12)=>13
([(4,10),(5,11),(6,11),(7,9),(7,10),(8,9),(8,11)],12)=>12
([(3,9),(3,10),(4,6),(4,7),(5,6),(5,7),(8,9),(8,10)],11)=>11
([(6,10),(7,9),(8,9),(8,10)],11)=>11
([(2,9),(3,9),(4,9),(5,9),(6,7),(7,8),(8,9)],10)=>11
([(0,12),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,12),(8,11),(9,11),(10,11)],13)=>17
([(7,10),(7,11),(8,9),(8,11),(9,11),(10,11)],12)=>12
([(8,11),(9,11),(10,11)],12)=>12
([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,7),(7,9),(8,10),(8,11),(9,10)],12)=>17
([(8,10),(8,11),(9,10),(9,11),(10,11)],12)=>12
([(1,10),(2,10),(3,10),(4,10),(5,6),(6,8),(7,9),(7,10),(8,9)],11)=>13
([(0,10),(1,9),(2,12),(3,11),(4,5),(4,6),(5,7),(6,11),(7,12),(8,9),(8,10)],13)=>18
([(0,11),(1,11),(2,10),(3,9),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10)],12)=>16
([(0,7),(1,6),(2,5),(3,4),(8,11),(9,10),(10,12),(11,12)],13)=>18
([(2,3),(4,11),(5,9),(6,8),(7,10),(8,9),(10,11)],12)=>14
([(0,8),(1,8),(2,7),(3,6),(4,5),(4,6),(5,7)],9)=>12
([(3,8),(4,7),(5,6),(5,7),(6,8)],9)=>9
([(1,7),(2,6),(3,5),(4,5),(6,8),(7,8)],9)=>10
([(2,8),(3,7),(4,6),(5,6),(7,8)],9)=>9
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6)],9)=>11
([(1,2),(3,8),(4,8),(5,7),(6,7)],9)=>10
([(1,8),(2,9),(3,9),(4,5),(4,6),(5,7),(6,8),(7,9)],10)=>12
([(2,9),(3,8),(4,5),(4,6),(5,7),(6,8),(7,9)],10)=>11
([(0,9),(1,9),(2,8),(3,7),(4,5),(4,6),(5,7),(6,8)],10)=>13
([(1,9),(2,9),(3,8),(4,7),(5,6),(5,7),(6,8)],10)=>12
([(2,3),(4,7),(5,6),(6,8),(7,9),(8,9)],10)=>11
([(4,9),(5,8),(6,7),(6,8),(7,9)],10)=>10
([(0,7),(1,6),(2,9),(3,8),(4,8),(4,9),(5,6),(5,7)],10)=>13
([(1,7),(2,6),(3,9),(4,8),(5,8),(5,9),(6,7)],10)=>12
([(0,1),(2,8),(3,7),(4,6),(5,6),(7,9),(8,9)],10)=>13
([(2,8),(3,7),(4,6),(5,6),(7,9),(8,9)],10)=>10
([(3,4),(5,8),(6,7),(7,9),(8,9)],10)=>10
([(2,9),(3,7),(4,6),(5,8),(6,7),(8,9)],10)=>11
([(0,7),(1,7),(2,6),(3,6),(4,9),(5,8),(8,9)],10)=>13
([(1,2),(3,9),(4,8),(5,7),(6,7),(8,9)],10)=>12
([(3,9),(4,8),(5,7),(6,7),(8,9)],10)=>10
([(1,9),(2,9),(3,8),(4,8),(5,7),(6,7)],10)=>11
([(0,3),(1,2),(4,9),(5,9),(6,8),(7,8)],10)=>13
([(4,9),(5,9),(6,8),(7,8)],10)=>10
([(3,6),(4,5),(7,9),(8,9)],10)=>10
([(1,10),(2,9),(3,4),(3,5),(4,6),(5,7),(6,8),(7,9),(8,10)],11)=>14
([(0,10),(1,11),(2,11),(3,11),(4,5),(4,6),(5,7),(6,8),(7,9),(8,10),(9,11)],12)=>16
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The Ramsey number of a graph.
This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1]
Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]
This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1]
Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]
References
[1] Chvatál, C., Rödl, V., Szemerédi, E., Trotter, Jr., W. T. The Ramsey number of a graph with bounded maximum degree MathSciNet:0714447
[2] Radziszowski, Stanisław P. Small Ramsey numbers MathSciNet:1670625
[3] wikipedia:Ramsey's theorem#Ramsey numbers
[4] Hendry, G. R. T. Ramsey numbers for graphs with five vertices MathSciNet:0994745
[5] Angeltveit, V., McKay, B. D. $R(5,5) \le 48$ arXiv:1703.08768
[2] Radziszowski, Stanisław P. Small Ramsey numbers MathSciNet:1670625
[3] wikipedia:Ramsey's theorem#Ramsey numbers
[4] Hendry, G. R. T. Ramsey numbers for graphs with five vertices MathSciNet:0994745
[5] Angeltveit, V., McKay, B. D. $R(5,5) \le 48$ arXiv:1703.08768
Code
N_vertices = 13 # the maximal number of vertices we consider
N_Ramsey = 7 # all graphs with Ramsey number at most N_Ramsey
statistic_dict = dict()
def statistic(G):
return statistic_dict.get(G.canonical_label().copy(immutable=True))
"""
The Ramsey number of a graph.
This is the smallest integer $n$ such that every two-colouring of the
$n$ vertices of the complete graph $K_n$ contains a (not necessarily
induced) monochromatic copy of the given graph. [1]
Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known. [2,3]
[1] Chvatál,Rödl,Szemerédi,Trotter, The Ramsey number of a graph with bounded maximum degree. doi:10.1016/0095-8956(83)90037-0
[2] Radziszowski, Stanisław P. "Small Ramsey numbers." Electron. J. Combin 1.7 (1994).
[3] [[wikipedia:Ramsey's theorem#Ramsey numbers]]
[4] Hendry, G. R. T. Ramsey numbers for graphs with five vertices
"""
def check_Ramsey(n, already_found=[]):
r"""
Colour the complete graph on n vertices with two colours, and
return the subgraphs which appear in all colourings.
EXAMPLES::
sage: L = dict()
sage: n = 2; L[n] = check_Ramsey(n)
sage: n = 3; L[n] = check_Ramsey(n, sum(L.values(), []))
sage: n = 4; L[n] = check_Ramsey(n, sum(L.values(), []))
sage: n = 5; L[n] = check_Ramsey(n, sum(L.values(), []))
sage: n = 6; L[n] = check_Ramsey(n, sum(L.values(), []))
sage: n = 7; L[n] = check_Ramsey(n, sum(L.values(), []))
sage: n=6; graphics_array([H.plot() for H in L[n] if H.is_connected()])
"""
def has_copy(H1, H2, H):
r"""
Return True if there is a copy of H in H1=s or H2=E\s.
"""
S1 = H1.subgraph_search(H)
if not S1 is None:
return True
S2 = H2.subgraph_search(H)
if not S2 is None:
return True
return False
candidates = [H.canonical_label() for k in range(n+1) for H in graphs(k)]
candidates = [H for H in candidates if H not in already_found]
G = graphs.CompleteGraph(n)
E = Set(G.edges(labels=False))
V = G.vertices()
for size in range(1+len(E)//2):
print("check subgraphs of size", size)
tested = []
for s in E.subsets(size):
H1 = Graph(n)
H1.add_edges(s)
H1 = H1.canonical_label()
H2 = Graph(n)
H2.add_edges(E.difference(s))
H2 = H2.canonical_label()
if (H1, H2) not in tested:
tested += [(H1, H2)]
candidates = [H for H in candidates if has_copy(H1, H2, H)]
return candidates
def add_to_dict(G, v):
H = G.canonical_label().copy(immutable=True)
if H in statistic_dict:
assert statistic_dict[H] == v, "The graph %s should have Ramsey number %s, got %s instead"%(repr(H), statistic_dict[H], v)
print("%s already known to have Ramsey value %s"%(repr(H), v))
else:
statistic_dict[H] = v
Ramsey_small = dict()
for n in range(N_Ramsey+1):
Ramsey_small[n] = check_Ramsey(n, sum(Ramsey_small.values(), []))
for v, lG in Ramsey_small.items():
for G in lG:
add_to_dict(G, v)
# table IIIa
Ramsey_almost_complete = dict()
Ramsey_almost_complete[3] = 3
Ramsey_almost_complete[4] = 10
Ramsey_almost_complete[5] = 22
for n, v in Ramsey_almost_complete.items():
G = graphs.CompleteGraph(n)
G.delete_edge(G.edges()[0])
add_to_dict(G, v)
# G8 - G20 are from Table 1 in
# Hendry, G. R. T. Ramsey numbers for graphs with five vertices
G8 = Graph([(1,2),(2,3),(3,4),(4,5),(3,5)]).canonical_label().copy(immutable=True)
G9 = Graph([(1,2),(2,3),(3,4),(4,5),(2,4)]).canonical_label().copy(immutable=True)
G10 = Graph([(1,2),(1,3),(1,4),(1,5),(2,3)]).canonical_label().copy(immutable=True)
G12 = Graph([(1,2),(2,3),(3,4),(4,5),(1,3),(3,5)]).canonical_label().copy(immutable=True)
G13 = Graph([(1,2),(1,3),(1,4),(1,5),(2,3),(3,4)]).canonical_label().copy(immutable=True)
G14 = Graph([(1,2),(1,3),(1,4),(2,5),(2,3),(3,4)]).canonical_label().copy(immutable=True)
G15 = Graph([(1,2),(2,3),(3,4),(4,5),(5,1),(1,3)]).canonical_label().copy(immutable=True)
G16 = Graph([(1,2),(2,3),(3,4),(4,5),(5,1),(1,3),(1,4)]).canonical_label().copy(immutable=True)
G17 = Graph([(1,2),(2,3),(3,4),(4,5),(5,1),(1,3),(2,4)]).canonical_label().copy(immutable=True)
G19 = Graph([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(1,5)]).canonical_label().copy(immutable=True)
G20 = Graph([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(1,5),(2,5)]).canonical_label().copy(immutable=True)
Ramsey_Hendry = [(G8, 9), (G9, 9), (G10, 9), (G12, 9), (G13, 10), (G14, 10), (G15, 9), (G16, 10), (G17, 10), (G19, 18), (G20, 18)]
for G, v in Ramsey_Hendry:
add_to_dict(G, v)
# table IVa and IVb
Ramsey_bipartite = dict()
Ramsey_bipartite[3, 3] = 18
Ramsey_bipartite[2, 2] = 6
Ramsey_bipartite[2, 3] = 10
Ramsey_bipartite[2, 4] = 14
Ramsey_bipartite[2, 5] = 18
Ramsey_bipartite[2, 6] = 21
Ramsey_bipartite[2, 7] = 26
Ramsey_bipartite[2, 8] = 30
Ramsey_bipartite[2, 9] = 33
Ramsey_bipartite[2, 10] = 38
Ramsey_bipartite[2, 11] = 42
Ramsey_bipartite[2, 12] = 46
Ramsey_bipartite[2, 13] = 50
Ramsey_bipartite[2, 14] = 54
Ramsey_bipartite[2, 15] = 57
Ramsey_bipartite[2, 16] = 62
for (n,m), v in Ramsey_bipartite.items():
add_to_dict(graphs.CompleteBipartiteGraph(n,m), v)
# section 3.3.2 a
def Ramsey_star(n):
m = n-1
assert m > 0
if is_even(m):
return 2*m-1
else:
return 2*m
for n in range(2, N_vertices+1):
add_to_dict(graphs.StarGraph(n-1), Ramsey_star(n))
# section 3.3.2, n
def Ramsey_m_4_star(n):
assert n % 4 == 0, "The number of vertices, %s, should be divisible by 4"%n
m = n//4
assert m >= 2, "There must be at least 2 copies of the star, but there are only %s"%m
return 5*m-1
for n in range(8, N_vertices+1, 4):
add_to_dict(sum([graphs.StarGraph(3) for k in range(n//4)], Graph()), Ramsey_m_4_star(n))
# section 4.1, a,b,c
def Ramsey_cycle(n):
assert n > 2
if is_odd(n) and n > 3:
return 2*n-1
elif is_even(n) and n > 4:
return n-1+(n//2)
elif n == 3 or n == 4:
return 6
for n in range(3, N_vertices+1):
add_to_dict(graphs.CycleGraph(n), Ramsey_cycle(n))
# section 4.1, e
def Ramsey_m_triangles(n):
assert n % 3 == 0, "The number of vertices, %s, should be divisible by 3"%n
m = n // 3
assert m >= 2, "There must be at least 2 copies of the triangle, but there are only %s"%m
return 5*m
for n in range(6, N_vertices+1, 3):
add_to_dict(sum([graphs.CycleGraph(3) for k in range(n//3)], Graph()), Ramsey_m_triangles(n))
# section 4.1, f
def Ramsey_m_squares(n):
assert n % 4 == 0, "The number of vertices, %s, should be divisible by 4"%n
m = n // 4
assert m >= 2, "There must be at least 2 copies of the square, but there are only %s"%m
return 6*m-1
for n in range(8, N_vertices+1, 4):
add_to_dict(sum([graphs.CycleGraph(4) for k in range(n//4)], Graph()), Ramsey_m_squares(n))
# section 5.1
def Ramsey_paths(m):
assert m >= 2
return m + (m//2) - 1
for n in range(2, N_vertices+1):
add_to_dict(graphs.PathGraph(n), Ramsey_paths(n))
# section 5.13 b
def Ramsey_union_K2(m):
assert is_even(m) and m >= 2
return 3*(m//2) - 1
for n in range(2, N_vertices+1, 2):
add_to_dict(sum([graphs.CompleteGraph(2) for k in range(n//2)], Graph()), Ramsey_union_K2(n))
Ramsey_wheel = dict()
# table VIII
Ramsey_wheel[3] = 6
Ramsey_wheel[4] = 18
Ramsey_wheel[5] = 15
Ramsey_wheel[6] = 17
for n, v in Ramsey_wheel.items():
add_to_dict(graphs.WheelGraph(n), v)
Ramsey_book = dict()
# table IX
Ramsey_book[1] = 6
Ramsey_book[2] = 10
Ramsey_book[3] = 14
Ramsey_book[4] = 18
Ramsey_book[5] = 21
Ramsey_book[6] = 26
for n, v in Ramsey_book.items():
add_to_dict(Graph(1).join(graphs.StarGraph(n)), v)
# R.J. Faudree and R.H. Schelp, Ramsey Numbers for All Linear Forests, Discrete Mathematics, 16 (1976) 149-155.
def Ramsey_linear_forest(n, j):
assert is_even(n-j)
return n + (n-j)//2 - 1
for n in range(2, N_vertices+1):
for la in Partitions(n):
if min(la) > 1:
G = sum([graphs.PathGraph(p) for p in la], Graph())
add_to_dict(G, Ramsey_linear_forest(n, sum(1 for p in la if is_odd(p))))
# J.W. Grossman, The Ramsey Numbers of the Union of Two Stars, Utilitas Mathematica, 16 (1979) 271-279.
def Ramsey_two_stars(n, m):
assert n >= m, "Ramsey_two_stars only valid for n >= m, but n = %s and m = %s"%(n,m)
return max(n+2*m, 2*n+1, n+m+3)
for n in range(2, N_vertices+1):
for m in range(2,(n-1)//2+1):
G = graphs.StarGraph(m-1) + graphs.StarGraph(n-m-1)
add_to_dict(G, Ramsey_two_stars(n-m-1, m-1))
# Yu, P., & Li, Y. (2016). All Ramsey numbers for brooms in graphs. The Electronic Journal of Combinatorics, 23(3), 3-29.
def Ramsey_broom(k, l):
assert k >= 2
n = k+l
if l == 1 or l == 2:
return Ramsey_star(n)
if l == 3:
return Ramsey_star(n-1)
if l >= 2*k - 1:
return n + (l+1)//2 - 1
if 4 <= l <= 2*k - 2:
return 2*n - 2*((l+1)//2) - 1
def BroomGraph(k, l):
G = graphs.StarGraph(k)
assert G.degree(0) == k
G.add_path([0] + [k+1+i for i in range(l-1)])
G.layout("tree", save_pos=True)
return G
for n in range(2, N_vertices+1):
for k in range(2,n-1):
G = BroomGraph(k, n-k)
add_to_dict(G, Ramsey_broom(k, n-k))
# Jerrold W. Grossman, Frank Harary, Maria Klawe, Generalized Ramsey theory for graphs, X: double stars
def Ramsey_double_star(k, l):
assert k >= 2
n = k+l+2
if is_odd(k) and l <= 2:
return max(2*k+1, k+2*l+2)
if (is_even(k) or l >= 3) and (k^2 <= 2*l or k >= 3*l):
return max(2*k+2, k+2*l+2)
def DoubleStarGraph(k, l):
assert k >= l, "DoubleStarGraph only defined for k >= l"
G = graphs.StarGraph(k)
H = G.disjoint_union(graphs.StarGraph(l), "pairs")
H.add_edge((0,0),(1,0))
return H
for n in range(2, N_vertices+1):
for l in range(2,n//2):
k = n-l-2
G = DoubleStarGraph(k, l)
assert G.num_verts() == n
if (is_odd(k) and l <= 2) or ((is_even(k) or l >= 3) and (k^2 <= 2*l or k >= 3*l)):
add_to_dict(G, Ramsey_double_star(k, l))
######################################################################
# finally, add isolated vertices
for G, v in list(statistic_dict.items()):
for k in range(N_vertices-G.num_verts()):
add_to_dict(G.disjoint_union(Graph(k)), max(G.num_verts()+k, v))
Created
May 07, 2016 at 08:53 by Martin Rubey
Updated
Jan 02, 2023 at 19:44 by Martin Rubey
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