Values
([],1) => ([],1) => 1
([],2) => ([],1) => 1
([(0,1)],2) => ([(0,1)],2) => 1
([],3) => ([],1) => 1
([(1,2)],3) => ([(0,1)],2) => 1
([(0,2),(1,2)],3) => ([(0,1)],2) => 1
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 2
([],4) => ([],1) => 1
([(2,3)],4) => ([(0,1)],2) => 1
([(1,3),(2,3)],4) => ([(0,1)],2) => 1
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => 1
([(0,3),(1,2)],4) => ([(0,1)],2) => 1
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => 1
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 2
([(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) => 1
([(2,4),(3,4)],5) => ([(0,1)],2) => 1
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 1
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 1
([(1,4),(2,3)],5) => ([(0,1)],2) => 1
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => 1
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2) => 1
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => 1
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2) => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => 1
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(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) => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 2
([(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) => 7
([],6) => ([],1) => 1
([(4,5)],6) => ([(0,1)],2) => 1
([(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(2,5),(3,4)],6) => ([(0,1)],2) => 1
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2) => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(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) => 2
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2) => 1
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2) => 1
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
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Description
The number of edges minus the number of vertices plus 2 of a graph.
When G is connected and planar, this is also the number of its faces.
When $G=(V,E)$ is a connected graph, this is its $k$-monochromatic index for $k>2$: for $2\leq k\leq |V|$, the $k$-monochromatic index of $G$ is the maximum number of edge colors allowed such that for each set $S$ of $k$ vertices, there exists a monochromatic tree in $G$ which contains all vertices from $S$. It is shown in [1] that for $k>2$, this is given by this statistic.
When G is connected and planar, this is also the number of its faces.
When $G=(V,E)$ is a connected graph, this is its $k$-monochromatic index for $k>2$: for $2\leq k\leq |V|$, the $k$-monochromatic index of $G$ is the maximum number of edge colors allowed such that for each set $S$ of $k$ vertices, there exists a monochromatic tree in $G$ which contains all vertices from $S$. It is shown in [1] that for $k>2$, this is given by this statistic.
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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