Values
([(0,1)],2) => ([],1) => ([],1) => 1
([(1,2)],3) => ([],1) => ([],1) => 1
([(0,2),(1,2)],3) => ([(0,1)],2) => ([(0,1)],2) => 2
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,3)],4) => ([],1) => ([],1) => 1
([(1,3),(2,3)],4) => ([(0,1)],2) => ([(0,1)],2) => 2
([(0,3),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(1,2)],4) => ([],2) => ([],1) => 1
([(0,3),(1,2),(2,3)],4) => ([(0,2),(1,2)],3) => ([(0,1)],2) => 2
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(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) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 2
([(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) => ([(0,1),(0,2),(1,2)],3) => 3
([(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) => ([(0,1),(0,2),(1,2)],3) => 3
([(3,4)],5) => ([],1) => ([],1) => 1
([(2,4),(3,4)],5) => ([(0,1)],2) => ([(0,1)],2) => 2
([(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,4),(2,4),(3,4)],5) => ([(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
([(1,4),(2,3)],5) => ([],2) => ([],1) => 1
([(1,4),(2,3),(3,4)],5) => ([(0,2),(1,2)],3) => ([(0,1)],2) => 2
([(0,1),(2,4),(3,4)],5) => ([(1,2)],3) => ([(0,1)],2) => 2
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 3
([(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) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(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) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(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),(1,3),(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) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => 2
([(0,1),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(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),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(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) => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(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) => ([(0,1),(0,2),(1,2)],3) => 3
([(4,5)],6) => ([],1) => ([],1) => 1
([(3,5),(4,5)],6) => ([(0,1)],2) => ([(0,1)],2) => 2
([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,5),(3,5),(4,5)],6) => ([(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
([(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) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
([(2,5),(3,4)],6) => ([],2) => ([],1) => 1
([(2,5),(3,4),(4,5)],6) => ([(0,2),(1,2)],3) => ([(0,1)],2) => 2
([(1,2),(3,5),(4,5)],6) => ([(1,2)],3) => ([(0,1)],2) => 2
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,1),(2,5),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(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
([(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) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(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) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,3),(1,2)],4) => ([(0,1)],2) => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 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) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(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) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(2,3)],6) => ([],3) => ([],1) => 1
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => 2
([(0,1),(2,5),(3,4),(4,5)],6) => ([(1,3),(2,3)],4) => ([(0,1)],2) => 2
([(1,2),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(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) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(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) => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => ([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 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) => ([(0,1),(0,2),(1,2)],3) => 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => ([(0,1)],2) => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 3
([(5,6)],7) => ([],1) => ([],1) => 1
([(4,6),(5,6)],7) => ([(0,1)],2) => ([(0,1)],2) => 2
([(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 3
([(2,6),(3,6),(4,6),(5,6)],7) => ([(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
([(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) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
([(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) => ([(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
([(3,6),(4,5)],7) => ([],2) => ([],1) => 1
([(3,6),(4,5),(5,6)],7) => ([(0,2),(1,2)],3) => ([(0,1)],2) => 2
([(2,3),(4,6),(5,6)],7) => ([(1,2)],3) => ([(0,1)],2) => 2
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Description
The Grundy number of a graph.
The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring.
In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ (St000098The chromatic number of a graph.), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ (St000171The degree of the graph.).
The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring.
In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ (St000098The chromatic number of a graph.), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ (St000171The degree of the graph.).
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 $G$.
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
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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