- FindStat

Identifier

Mp00198: Posets —incomparability graph⟶ Graphs
St000098: Graphs ⟶ ℤ

Values

([],1) => ([],1) => 1

([],2) => ([(0,1)],2) => 2

([(0,1)],2) => ([],2) => 1

([],3) => ([(0,1),(0,2),(1,2)],3) => 3

([(1,2)],3) => ([(0,2),(1,2)],3) => 2

([(0,1),(0,2)],3) => ([(1,2)],3) => 2

([(0,2),(2,1)],3) => ([],3) => 1

([(0,2),(1,2)],3) => ([(1,2)],3) => 2

([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4

([(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3

([(1,2),(1,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3

([(0,1),(0,2),(0,3)],4) => ([(1,2),(1,3),(2,3)],4) => 3

([(0,2),(0,3),(3,1)],4) => ([(1,3),(2,3)],4) => 2

([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => 2

([(1,2),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 2

([(0,3),(3,1),(3,2)],4) => ([(2,3)],4) => 2

([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 3

([(0,3),(1,3),(3,2)],4) => ([(2,3)],4) => 2

([(0,3),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => 3

([(0,3),(1,2)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 2

([(0,3),(1,2),(1,3)],4) => ([(0,3),(1,2),(2,3)],4) => 2

([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,3),(1,2)],4) => 2

([(0,3),(2,1),(3,2)],4) => ([],4) => 1

([(0,3),(1,2),(2,3)],4) => ([(1,3),(2,3)],4) => 2

([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 5

([(3,4)],5) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

([(2,3),(2,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

([(1,2),(1,3),(1,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

([(0,1),(0,2),(0,3),(0,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

([(0,2),(0,3),(0,4),(4,1)],5) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => ([(2,3),(2,4),(3,4)],5) => 3

([(1,3),(1,4),(4,2)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,3),(0,4),(4,1),(4,2)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => 3

([(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(3,4)],5) => 2

([(0,3),(0,4),(3,2),(4,1)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 2

([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(1,4),(2,3)],5) => 2

([(2,3),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(1,4),(4,2),(4,3)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(4,1),(4,2),(4,3)],5) => ([(2,3),(2,4),(3,4)],5) => 3

([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

([(1,4),(2,4),(4,3)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,4),(4,2),(4,3)],5) => ([(1,4),(2,3)],5) => 2

([(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

([(0,4),(1,4),(2,4),(4,3)],5) => ([(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,4),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

([(0,4),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 3

([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 3

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,4),(2,3),(4,2)],5) => ([(3,4)],5) => 2

([(0,4),(1,3),(2,3),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,4),(2,3),(2,4)],5) => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,4),(2,3),(3,4)],5) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(1,4),(2,3)],5) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 3

([(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,2),(1,4),(2,3)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2

([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 2

([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 3

([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => ([(0,1),(2,4),(3,4)],5) => 2

([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => ([(1,4),(2,3)],5) => 2

([(0,4),(1,2),(1,4),(4,3)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

([(0,4),(1,2),(1,3)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 3

([(0,4),(1,2),(1,3),(1,4)],5) => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 3

([(0,2),(0,4),(3,1),(4,3)],5) => ([(1,4),(2,4),(3,4)],5) => 2

([(0,4),(1,2),(1,3),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 3

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(2,4),(3,4)],5) => 2

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,3),(0,4),(1,2),(1,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 3

([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 3

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => ([(0,1),(2,3),(2,4),(3,4)],5) => 3

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2

([(0,3),(1,2),(1,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2

([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => ([(0,1),(2,4),(3,4)],5) => 2

([(1,4),(3,2),(4,3)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 2

([(0,3),(3,4),(4,1),(4,2)],5) => ([(3,4)],5) => 2

([(1,4),(2,3),(3,4)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,2),(2,4),(4,3)],5) => ([(2,4),(3,4)],5) => 2

([(0,3),(1,4),(4,2)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,4),(3,2),(4,1),(4,3)],5) => ([(2,4),(3,4)],5) => 2

([(0,4),(1,2),(2,3),(2,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 2

([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => 1

([(0,3),(1,2),(2,4),(3,4)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,4),(1,2),(2,3),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => 2

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => 2

([],6) => ([(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) => 6

([(4,5)],6) => ([(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) => 5

([(3,4),(3,5)],6) => ([(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) => 5

([(2,3),(2,4),(2,5)],6) => ([(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) => 5

([(1,2),(1,3),(1,4),(1,5)],6) => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5

([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

([(1,3),(1,4),(1,5),(5,2)],6) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

([(1,2),(1,3),(1,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4

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$F_{1} = q$

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

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

$F_{4} = q + 9\ q^{2} + 5\ q^{3} + q^{4}$

$F_{5} = q + 25\ q^{2} + 29\ q^{3} + 7\ q^{4} + q^{5}$

$F_{6} = q + 74\ q^{2} + 170\ q^{3} + 63\ q^{4} + 9\ q^{5} + q^{6}$

Description

The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Map

incomparability graph

Description

The incomparability graph of a poset.