***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000450 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- 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 ----------------------------------------------------------------------------- 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 ----------------------------------------------------------------------------- Created: Mar 29, 2016 at 11:12 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Mar 30, 2016 at 09:36 by Christian Stump