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Statistic identifier: St000450
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Collection: Graphs
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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.
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References: [1] Li, X., Wu, D. The (vertex-)monochromatic index of a graph [[arXiv:1603.05338]]
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Code:
def statistic(G):
return len(G.edges())-len(G.vertices())+2
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Statistic values:
([],1) => 1
([],2) => 0
([(0,1)],2) => 1
([],3) => -1
([(1,2)],3) => 0
([(0,2),(1,2)],3) => 1
([(0,1),(0,2),(1,2)],3) => 2
([],4) => -2
([(2,3)],4) => -1
([(1,3),(2,3)],4) => 0
([(0,3),(1,3),(2,3)],4) => 1
([(0,3),(1,2)],4) => 0
([(0,3),(1,2),(2,3)],4) => 1
([(1,2),(1,3),(2,3)],4) => 1
([(0,3),(1,2),(1,3),(2,3)],4) => 2
([(0,2),(0,3),(1,2),(1,3)],4) => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([],5) => -3
([(3,4)],5) => -2
([(2,4),(3,4)],5) => -1
([(1,4),(2,4),(3,4)],5) => 0
([(0,4),(1,4),(2,4),(3,4)],5) => 1
([(1,4),(2,3)],5) => -1
([(1,4),(2,3),(3,4)],5) => 0
([(0,1),(2,4),(3,4)],5) => 0
([(2,3),(2,4),(3,4)],5) => 0
([(0,4),(1,4),(2,3),(3,4)],5) => 1
([(1,4),(2,3),(2,4),(3,4)],5) => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
([(1,3),(1,4),(2,3),(2,4)],5) => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
([(0,4),(1,3),(2,3),(2,4)],5) => 1
([(0,1),(2,3),(2,4),(3,4)],5) => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 3
([(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) => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7
([],6) => -4
([(4,5)],6) => -3
([(3,5),(4,5)],6) => -2
([(2,5),(3,5),(4,5)],6) => -1
([(1,5),(2,5),(3,5),(4,5)],6) => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1
([(2,5),(3,4)],6) => -2
([(2,5),(3,4),(4,5)],6) => -1
([(1,2),(3,5),(4,5)],6) => -1
([(3,4),(3,5),(4,5)],6) => -1
([(1,5),(2,5),(3,4),(4,5)],6) => 0
([(0,1),(2,5),(3,5),(4,5)],6) => 0
([(2,5),(3,4),(3,5),(4,5)],6) => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
([(2,4),(2,5),(3,4),(3,5)],6) => 0
([(0,5),(1,5),(2,4),(3,4)],6) => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,4),(2,3)],6) => -1
([(1,5),(2,4),(3,4),(3,5)],6) => 0
([(0,1),(2,5),(3,4),(4,5)],6) => 0
([(1,2),(3,4),(3,5),(4,5)],6) => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 3
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 4
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 4
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 4
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 3
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 3
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 3
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 3
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 4
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 4
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 4
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 4
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 5
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 7
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 2
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 4
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 5
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 4
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 4
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 6
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 5
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 6
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],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) => 8
([(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) => 9
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 7
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 7
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 7
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 8
([(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) => 8
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9
([(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) => 10
([(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) => 11
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Created: Mar 29, 2016 at 11:12 by Christian Stump
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Last Updated: Mar 30, 2016 at 09:36 by Christian Stump