- FindStat

Identifier

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001117: Graphs ⟶ ℤ

Values

['A',1] => ([],1) => ([],1) => ([],1) => 0
['A',2] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => ([(1,2)],3) => 1
['B',2] => ([(0,3),(1,3),(3,2)],4) => ([(0,3),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => 3
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5
['A',3] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

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$F_{1} = 1$

$F_{2} = q + q^{3} + q^{5}$

Description

The game chromatic index of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored edge of the graph. Alice begins. If it is not possible for either player to colour a edge, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic index is the smallest number of colours such that Alice has a winning strategy.

Map

to graph

Description

Returns the Hasse diagram of the poset as an undirected graph.

Map

to root poset

Description

The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where

$\alpha \prec \beta$ if

$\beta - \alpha$ is a simple root.

Map

complement

Description

The complement of a graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.