Identifier
Values
[1,1] => ([(0,1)],2) => ([],1) => ([],1) => 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 1
[1,2] => ([(1,2)],3) => ([],1) => ([],1) => 1
[2,1] => ([(0,2),(1,2)],3) => ([(0,1)],2) => ([],2) => 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(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) => ([(0,5),(1,4),(2,3)],6) => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(2,3)],4) => 2
[1,3] => ([(2,3)],4) => ([],1) => ([],1) => 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3)],5) => 1
[2,2] => ([(1,3),(2,3)],4) => ([(0,1)],2) => ([],2) => 1
[3,1] => ([(0,3),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(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) => ([(0,5),(1,4),(2,3)],6) => 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(2,3)],4) => 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => 2
[1,4] => ([(3,4)],5) => ([],1) => ([],1) => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3)],5) => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
[2,3] => ([(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => 1
[3,2] => ([(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],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) => ([(0,5),(1,4),(2,3)],6) => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(2,3)],4) => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],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) => ([(2,5),(3,5),(4,5)],6) => 3
[1,5] => ([(4,5)],6) => ([],1) => ([],1) => 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3)],5) => 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => 2
[2,4] => ([(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => 1
[3,3] => ([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],5) => 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(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) => ([(0,5),(1,4),(2,3)],6) => 1
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(2,3)],4) => 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => 2
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(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) => ([(2,5),(3,5),(4,5)],6) => 3
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,6),(3,6),(4,6),(5,6)],7) => 3
[1,6] => ([(5,6)],7) => ([],1) => ([],1) => 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3)],5) => 1
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => 2
[2,5] => ([(4,6),(5,6)],7) => ([(0,1)],2) => ([],2) => 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => 1
[3,4] => ([(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],5) => 1
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(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) => 1
search for individual values
searching the database for the individual values of this statistic
Description
The number of distinct colouring schemes of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the number of distinct partitions that occur.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$.
Map
line graph
Description
The line graph of a graph.
Let $G$ be a graph with edge set $E$. Then its line graph is the graph with vertex set $E$, such that two vertices $e$ and $f$ are adjacent if and only if they are incident to a common vertex in $G$.
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.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.