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Identifier
Values
=>
Cc0020;cc-rep
([],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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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.
References
[1] Li, X., Wu, D. The (vertex-)monochromatic index of a graph arXiv:1603.05338
Code
def statistic(G):
    return len(G.edges())-len(G.vertices())+2
Created
Mar 29, 2016 at 11:12 by Christian Stump
Updated
Mar 30, 2016 at 09:36 by Christian Stump