Values
([],1) => ([],1) => 1
([],2) => ([],1) => 1
([(0,1)],2) => ([(0,1)],2) => 2
([],3) => ([],1) => 1
([(1,2)],3) => ([(0,1)],2) => 2
([(0,2),(1,2)],3) => ([(0,1)],2) => 2
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 3
([],4) => ([],1) => 1
([(2,3)],4) => ([(0,1)],2) => 2
([(1,3),(2,3)],4) => ([(0,1)],2) => 2
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => 2
([(0,3),(1,2)],4) => ([(0,1)],2) => 2
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => 2
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([],5) => ([],1) => 1
([(3,4)],5) => ([(0,1)],2) => 2
([(2,4),(3,4)],5) => ([(0,1)],2) => 2
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 2
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 2
([(1,4),(2,3)],5) => ([(0,1)],2) => 2
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => 2
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2) => 2
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => 2
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2) => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => 2
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],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) => 1
([(4,5)],6) => ([(0,1)],2) => 2
([(3,5),(4,5)],6) => ([(0,1)],2) => 2
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 2
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 2
([(2,5),(3,4)],6) => ([(0,1)],2) => 2
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 2
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2) => 2
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 2
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 2
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2) => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2) => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2) => 2
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2) => 2
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 2
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2) => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 3
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Description
The distinguishing number of a graph.
This is the minimal number of colours needed to colour the vertices of a graph, such that only the trivial automorphism of the graph preserves the colouring.
For connected graphs, this statistic is at most one plus the maximal degree of the graph, with equality attained for complete graphs, complete bipartite graphs and the cycle with five vertices, see Theorem 4.2 of [2].
This is the minimal number of colours needed to colour the vertices of a graph, such that only the trivial automorphism of the graph preserves the colouring.
For connected graphs, this statistic is at most one plus the maximal degree of the graph, with equality attained for complete graphs, complete bipartite graphs and the cycle with five vertices, see Theorem 4.2 of [2].
Map
core
Description
The core of a graph.
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
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