Values
=>
Cc0020;cc-rep
([],1)=>1
([],2)=>1
([(0,1)],2)=>2
([],3)=>1
([(1,2)],3)=>2
([(0,2),(1,2)],3)=>3
([(0,1),(0,2),(1,2)],3)=>3
([],4)=>1
([(2,3)],4)=>2
([(1,3),(2,3)],4)=>3
([(0,3),(1,3),(2,3)],4)=>4
([(0,3),(1,2)],4)=>3
([(0,3),(1,2),(2,3)],4)=>3
([(1,2),(1,3),(2,3)],4)=>3
([(0,3),(1,2),(1,3),(2,3)],4)=>4
([(0,2),(0,3),(1,2),(1,3)],4)=>4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>4
([],5)=>1
([(3,4)],5)=>2
([(2,4),(3,4)],5)=>3
([(1,4),(2,4),(3,4)],5)=>4
([(0,4),(1,4),(2,4),(3,4)],5)=>5
([(1,4),(2,3)],5)=>3
([(1,4),(2,3),(3,4)],5)=>3
([(0,1),(2,4),(3,4)],5)=>3
([(2,3),(2,4),(3,4)],5)=>3
([(0,4),(1,4),(2,3),(3,4)],5)=>4
([(1,4),(2,3),(2,4),(3,4)],5)=>4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>5
([(1,3),(1,4),(2,3),(2,4)],5)=>4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)=>4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)=>5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>5
([(0,4),(1,3),(2,3),(2,4)],5)=>4
([(0,1),(2,3),(2,4),(3,4)],5)=>4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)=>5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)=>5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)=>5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)=>4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)=>5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)=>5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>5
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>5
([],6)=>1
([(4,5)],6)=>2
([(3,5),(4,5)],6)=>3
([(2,5),(3,5),(4,5)],6)=>4
([(1,5),(2,5),(3,5),(4,5)],6)=>5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>6
([(2,5),(3,4)],6)=>3
([(2,5),(3,4),(4,5)],6)=>3
([(1,2),(3,5),(4,5)],6)=>3
([(3,4),(3,5),(4,5)],6)=>3
([(1,5),(2,5),(3,4),(4,5)],6)=>4
([(0,1),(2,5),(3,5),(4,5)],6)=>4
([(2,5),(3,4),(3,5),(4,5)],6)=>4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>5
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(2,4),(2,5),(3,4),(3,5)],6)=>4
([(0,5),(1,5),(2,4),(3,4)],6)=>4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>5
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>5
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>5
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>5
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,5),(1,4),(2,3)],6)=>3
([(1,5),(2,4),(3,4),(3,5)],6)=>4
([(0,1),(2,5),(3,4),(4,5)],6)=>4
([(1,2),(3,4),(3,5),(4,5)],6)=>4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>4
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)=>4
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>5
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>5
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>6
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>5
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>4
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)=>5
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>4
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>4
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)=>4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)=>4
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)=>5
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)=>5
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)=>4
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>4
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)=>5
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>5
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>5
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)=>6
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>5
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>5
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>5
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>5
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>5
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>5
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)=>5
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)=>5
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>5
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>5
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>5
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>5
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)=>5
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>5
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)=>6
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)=>5
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)=>5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)=>5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>5
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)=>5
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>6
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)=>6
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>6
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>6
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>5
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>6
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>6
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6
([],7)=>1
([(5,6)],7)=>2
([(4,6),(5,6)],7)=>3
([(3,6),(4,6),(5,6)],7)=>4
([(2,6),(3,6),(4,6),(5,6)],7)=>5
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>7
([(3,6),(4,5)],7)=>3
([(3,6),(4,5),(5,6)],7)=>3
([(2,3),(4,6),(5,6)],7)=>3
([(4,5),(4,6),(5,6)],7)=>3
([(2,6),(3,6),(4,5),(5,6)],7)=>4
([(1,2),(3,6),(4,6),(5,6)],7)=>4
([(3,6),(4,5),(4,6),(5,6)],7)=>4
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>5
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)=>5
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>6
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(3,5),(3,6),(4,5),(4,6)],7)=>4
([(1,6),(2,6),(3,5),(4,5)],7)=>4
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)=>4
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>4
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)=>4
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>4
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>4
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>5
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>5
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>5
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)=>5
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)=>5
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>5
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)=>5
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>7
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(1,6),(2,5),(3,4)],7)=>3
([(2,6),(3,5),(4,5),(4,6)],7)=>4
([(1,2),(3,6),(4,5),(5,6)],7)=>4
([(0,3),(1,2),(4,6),(5,6)],7)=>4
([(2,3),(4,5),(4,6),(5,6)],7)=>4
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>4
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)=>4
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)=>4
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)=>4
([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>5
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>5
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>7
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)=>5
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)=>4
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>4
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>5
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)=>5
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>4
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)=>4
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)=>5
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)=>5
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)=>5
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)=>4
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)=>4
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)=>4
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)=>4
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)=>5
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>4
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>4
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)=>5
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)=>4
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)=>4
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4
([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>4
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>4
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)=>4
([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)=>5
([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)=>5
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)=>5
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)=>5
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)=>5
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)=>6
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)=>6
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)=>4
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)=>5
([(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>5
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>6
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)=>5
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>5
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)=>6
([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)=>5
([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>5
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)=>6
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)=>5
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>5
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7)=>5
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)=>6
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)=>5
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)=>5
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)=>6
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)=>5
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)=>5
([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>5
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>6
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>7
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>6
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)=>6
([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>7
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Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
References
Code
def statistic(G):
G = G.canonical_label().copy(immutable=True)
return statistic_aux(G)
@cached_function
def statistic_aux(G):
"""
sage: N = 8; lG = [G for n in range(1, N) for G in graphs(n)]
sage: all(statistic(G) >= max(G.degree()) + 1 for G in lG)
True
sage: all(statistic(G) <= 2*max(G.degree())*sqrt(G.num_verts()-1) for G in lG if G.edges())
True
sage: all((statistic(G) == G.num_verts()) == (G.diameter() <= 2) for G in lG)
True
sage: all(statistic(graphs.CompleteGraph(n)) == n for n in range(6))
True
sage: def min_k(G):
....: k = 0
....: while True:
....: if binomial(k, 2) >= G.num_edges():
....: return k
....: k += 1
sage: def harmonious_path(n):
....: k = min_k(graphs.PathGraph(n))
....: if is_odd(k) or (k-2)//2 <= binomial(k, 2) - n + 1 <= k-2:
....: return k
....: return k+1
sage: all(statistic(graphs.PathGraph(n)) == harmonious_path(n) for n in range(1, 8))
True
"""
G = G.relabel(inplace=False)
n = G.num_verts()
K = range(n) # colors
Kp = [(i, j) for i in K for j in range(i+1, n)] # pairs of colours
V = G.vertices()
E = [tuple(sorted(e)) for e in G.edges(labels=False)]
P = MixedIntegerLinearProgram(maximization=False)
# y[c] == 1 if c is used
y = P.new_variable(binary=True, indices=K)
# x[(v,c)] == 1 if v is colored with c
x = P.new_variable(binary=True, indices=cartesian_product([V, K]))
# z[e, c, d] == 1 if the edge e is incident to colours c < d
z = P.new_variable(binary=True, indices=cartesian_product([E, Kp]))
P.set_objective(sum(y[c] for c in K))
for v in V:
# one color per node
P.add_constraint(sum(x[(v,c)] for c in K) == 1)
for c in K:
# if vertex v takes color c, activate y[c]
P.add_constraint(x[(v,c)] <= y[c])
for u, v in E:
for c in K:
# the colouring is proper
P.add_constraint(x[(u,c)] + x[(v,c)] <= 1)
if E:
for c, d in Kp:
P.add_constraint(x[(u,c)] + x[(v,d)] <= 1 + z[((u, v), (c, d))])
P.add_constraint(x[(u,d)] + x[(v,c)] <= 1 + z[((u, v), (c, d))])
if E:
for c, d in Kp:
P.add_constraint(sum(z[((u, v), (c, d))] for u, v in E) <= 1)
return ZZ(P.solve())
Created
Jun 03, 2021 at 09:53 by Martin Rubey
Updated
Jun 03, 2021 at 09:53 by Martin Rubey
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