- FindStat

Identifier

Mp00250: Graphs —clique graph⟶ Graphs
St000323: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

([],4) => ([],4) => 0

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

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

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

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

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

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

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

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

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

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

([],5) => ([],5) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => ([],6) => 0

([(4,5)],6) => ([],5) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => 0

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4$

$F_{4} = 11$

$F_{5} = 34$

Description

The minimal crossing number of a graph.
A drawing of a graph

$G$ is a drawing in

$\mathbb{R}^2$ such that

the vertices of $G$ are distinct points,
the edges of $G$ are simple curves joining their endpoints,
no edge passes through a vertex, and
no three edges cross in a common point.

The minimal crossing number of

$G$ is then the minimal number of crossings of edges in a drawing of

$G$ .
In particular, a graph is planar if and only if its minimal crossing number is

$0$ .
It is moreover conjectured that the crossing number of the complete graph

$K_n$ [1] is

$\frac{1}{4}\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{n-2}{2} \rfloor\lfloor \frac{n-3}{2} \rfloor,$
and the crossing number of the complete bipartite graph

$K_{n,m}$ [2] is

$\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{m}{2} \rfloor\lfloor \frac{m-1}{2} \rfloor.$
A general algorithm to compute the crossing number is e.g. given in [3].
This statistics data was provided by Markus Chimani [6].

Map

clique graph

Description

The clique graph of a graph.
The clique graph of a graph

$G$ has as vertex set the set of maximal cliques

$G$ and an edge between vertices corresponding to cliques that intersect.
In other words, it is the intersection graph of the maximal cliques of

$G$ .