Identifier
Values
[[],[[[[]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[[[]],[[[]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[[[[]]],[[]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[[[[[]]]],[]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[[[[[[]]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[[],[],[[[[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[],[[]],[[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[],[[[]]],[[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[],[[[[]]]],[]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[],[[],[[[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[],[[[[]]],[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[],[[[],[[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[],[[[[]],[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[],[[[[],[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[],[[[[[]]]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => 5
[[[]],[],[[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[]],[[[]]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[]],[[],[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[]],[[[]],[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[]],[[[],[]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[[]],[[[[]]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => 5
[[[[]]],[],[[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[]]],[[]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[],[]],[[[]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[[[]]],[[],[]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[[[]]],[[[]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => 5
[[[[[]]]],[],[]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[[],[[]]],[[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[]],[]],[[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[],[]]],[[]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[[[[]]]],[[]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => 5
[[[],[[[]]]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[[]]],[]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[],[[]]]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[[]],[]]],[]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[[],[]]]],[]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[[[[[]]]]],[]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => 5
[[[],[[[[]]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[[[]],[[[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[[]]],[[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[[[]]]],[]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[[[],[[[]]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[[[]]],[]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[[],[[]]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[[[]],[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => ([(0,4),(0,5),(1,2),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7) => 4
[[[[[[],[]]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 4
[[[[[[[]]]]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => ([(0,3),(0,6),(1,2),(1,5),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => 5
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Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges.
Map
complement
Description
The complement of a graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.
Map
square
Description
The square of a graph.
For a graph $G$, the square is the graph on the same set of vertices where two vertices are joined by an edge if there is a path in $G$ of length at most two between the two.
In other words, a vertex gets joint to its $2$-neighbourhood in G.
For a graph $G$, the square is the graph on the same set of vertices where two vertices are joined by an edge if there is a path in $G$ of length at most two between the two.
In other words, a vertex gets joint to its $2$-neighbourhood in G.
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