LGI2PResearch Center

Coloration des graphes de reines

michel.vasquez@mines-ales.fr

LGI2P Ecole des Mines d’Alès

2

Outline

About the Queen Graph Coloring Problem Definition Conjecture ?

A Complete Algorithm Reformulation of the coloring problem Efficient filtering

A Geometric Based Heuristic Geometric Operators Results synthesis

Coloring Extension

3

Rule for moving the queen on the chessboard

A

Each queen controls:• 1 column• 1 row• 2 diagonals

4

Graph definition

1 square of the chessboard vertex

2 squares controlled by the same queen edge

5

Graph definition: from chessboard to queen graph

a queen graph instance

G(V,E) with :V n2 vertices and E n3 edges

6

The Queen Graph Coloring Problem: definition

Given a chessboard,

what is the minimum number of colors required to cover it without clash between two queens of the same color ?

7

The Queen Graph Coloring Problem: what we know

The chromatic number of Queen-72 is 7 : (7) 7

(and (n) n if n is prime with 2 and 3)

1 2 3 4 5 6 74 5 6 7 1 2 37 1 2 3 4 5 63 4 5 6 7 1 26 7 1 2 3 4 52 3 4 5 6 7 15 6 7 1 2 3 4

8

Conjecture ?

The chromatic number of the Queen Graph is equal to n if and only if n is prime with 2 and 3

M. Gardner, 1969 : The Unexpected Hanging and Other Mathematical Diversions, Simon and Schuster, New York.

9

Conjecture ?

The chromatic number of the Queen Graph is equal to n if and only if n is prime with 2 and 3

E. Y. Gik, 1983 : Shakhmaty i matematika, Bibliotechka Kvant, vol. 24, Nauka, Moscow.

The chromatic number of the Queen Graph is equal to n if and only if n is prime with 2 and 3

10

Intox…

11

Intox…

12

Until 2003 no result are available for the queen graph chromatic number

when n is greater than 9 and n is multiple of 2 or 3

13

Outline

About the Queen Graph Coloring Problem

A Complete Algorithm Reformulation of the coloring problem Efficient filtering

A Geometric Based Heuristic Geometric Operators Results synthesis

Coloring Extension

14

Property (1)

The n rows, the n columns and the 2 main diagonals

are cliques with n vertices of the Queen-n2 graphÞ (n) n

15

Question (1)

For a given n, is (n) equal to n ?

saying it differently

Is there a partition of the Queen-n2 graph in n independent sets ?

16

Property (2)

A stable set cannot contain more than n vertices

To answer yes to question (1) and cover nn squares : each independent set must contain at least n vertices

AA

AA

AA

A

17

Question (2)

Are there n independent sets with exactly n vertices which do not cover themselves ?

18

General Algorithm

Step 1) Enumerate the independent sets with n vertices

(n queens that do not attack themselves)

Step 2) Find n among them which do not intersect

(solve the CSP)

19

Avoiding many equivalent coloring permutations

n squares belonging to a same clique are colored once for all:

1 2 3 4 5 6 7

20

Computing IS by backtracking

Enumeration : backtracking

n |V| |E| |I.S.| sec.10 100 1470 724 011 121 1980 2680 012 144 2596 14200 013 169 3328 73712 214 196 4186 365596 1515 225 5180 2279184 73

21

A CSP with n variables (corresponding to a n squares)

Spreading of the independent sets for Queen-102

56 36 48 69 63 63 69 48 36 56 176 946 839 785 046 00036 52 67 51 66 66 51 67 52 3648 67 66 43 48 48 43 66 67 4869 51 43 56 53 53 56 43 51 6963 66 48 53 42 42 53 48 66 6363 66 48 53 42 42 53 48 66 6369 51 43 56 53 53 56 43 51 6948 67 66 43 48 48 43 66 67 4836 52 67 51 66 66 51 67 52 3656 36 48 69 63 63 69 48 36 56

22

21 24 31 31 25 27 21 14 27 2 595 187 803 60013 21 32 34 31 27 28 21 1423 33 33 21 19 28 18 27 1929 24 25 18 24 24 21 25 3130 25 14 29 15 21 29 23 3533 25 13 28 22 15 16 34 3533 27 19 17 25 17 29 23 31

19 39 19 24 20 19 30 32 1917 28 25 21 33 28 31 22 1622 19 22 30 27 33 28 15 25

Branching on the smallest domain variable

Non overlapping constraints propagation

12 15 17 14 11 9 6 18 458 045 280 15 13 19 17 19 9 11 5

12 19 15 11 13 13 11 1016 12 14 8 8 15 12 2017 12 4 16 11 11 17 2117 16 11 13 7 8 11 20 12 16 10 8 17 9 14 19 5 18 15 11 11 16 19 86 18 10 13 17 20 11 916 9 13 16 17 11 8 12

The search space size is decreasing geometrically

23

First result

n = 10 : no solution 7000 seconds Þ (10) = 11

24

Filtering (principle)

Consider the cliques of the graph constituted by the uncolored vertices

If such a clique contains k vertices then you need at least k colors (i.e. k independent sets) to complete the process

25

Efficient Filtering (computationally)

Diagonals constitute cliques (and are easy to handle):

Þ for a given diagonal there is at most one vertex that can come from a specific stable set,

Þ at level k of the search tree, diagonals must contain less than n-k empty squares

Delete all the independent sets that do not verify this condition

26

Efficient Filtering (experimentally)

At the root of the search tree

AA

AA

AA

A

this independent set is excluded from the search space

27

Efficient Filtering (experimentally)

Search space reduction

n |V| |E| |I.S.| |I.S.| f iltered10 100 1470 724 54411 121 1980 2680 174412 144 2596 14200 944013 169 3328 73712 5200814 196 4186 365596 23808815 225 5180 2279184 1484400

28

Efficient Filtering (experimentally)

At each level : 4 more constraints

29

First Results : complete method

(10) no solution 1 second(maximum depth of backtrack in the search tree : 5 rather than 10)

(12) 12 454 solutions 6963 seconds(exhaustive search)

(14) 14 1 solution en 142 hours(search aborted after one week)

30

Interest of filtering

Comparative results on n=12

31

Outline

About the Queen Graph Coloring Problem Definition Intox/Conjecture ?

A Complete Algorithm Reformulation of the coloring problem Efficient filtering

A Geometric Based Heuristic Geometric Operators Results synthesis

Coloring Extension

32

Certificate for n = 12

1 5 7 9 10 3 2 11 12 6 8 411 8 1 6 2 9 12 3 7 4 5 109 2 11 8 4 6 7 1 5 10 3 123 4 10 12 7 8 5 6 9 11 1 27 12 3 4 5 11 10 8 1 2 9 66 10 5 1 12 2 3 9 4 8 11 75 9 6 2 11 1 4 10 3 7 12 88 11 4 3 6 12 9 7 2 1 10 54 3 9 11 8 7 6 5 10 12 2 1

10 1 12 7 3 5 8 2 6 9 4 1112 7 2 5 1 10 11 4 8 3 6 92 6 8 10 9 4 1 12 11 5 7 3

33

Certificate for n = 12

1 5 7 9 10 3 2 11 12 6 8 411 8 1 6 2 9 12 3 7 4 5 109 2 11 8 4 6 7 1 5 10 3 123 4 10 12 7 8 5 6 9 11 1 27 12 3 4 5 11 10 8 1 2 9 66 10 5 1 12 2 3 9 4 8 11 75 9 6 2 11 1 4 10 3 7 12 88 11 4 3 6 12 9 7 2 1 10 54 3 9 11 8 7 6 5 10 12 2 1

10 1 12 7 3 5 8 2 6 9 4 1112 7 2 5 1 10 11 4 8 3 6 92 6 8 10 9 4 1 12 11 5 7 3

34

Certificate for n = 12

1 5 7 9 10 3 2 11 12 6 8 411 8 1 6 2 9 12 3 7 4 5 109 2 11 8 4 6 7 1 5 10 3 123 4 10 12 7 8 5 6 9 11 1 27 12 3 4 5 11 10 8 1 2 9 66 10 5 1 12 2 3 9 4 8 11 75 9 6 2 11 1 4 10 3 7 12 88 11 4 3 6 12 9 7 2 1 10 54 3 9 11 8 7 6 5 10 12 2 1

10 1 12 7 3 5 8 2 6 9 4 1112 7 2 5 1 10 11 4 8 3 6 92 6 8 10 9 4 1 12 11 5 7 3

35

Certificate for n = 12

1 5 7 9 10 3 2 11 12 6 8 411 8 1 6 2 9 12 3 7 4 5 109 2 11 8 4 6 7 1 5 10 3 123 4 10 12 7 8 5 6 9 11 1 27 12 3 4 5 11 10 8 1 2 9 66 10 5 1 12 2 3 9 4 8 11 75 9 6 2 11 1 4 10 3 7 12 88 11 4 3 6 12 9 7 2 1 10 54 3 9 11 8 7 6 5 10 12 2 1

10 1 12 7 3 5 8 2 6 9 4 1112 7 2 5 1 10 11 4 8 3 6 92 6 8 10 9 4 1 12 11 5 7 3

36

Exact but incomplete method

Assumption on the distribution of the colors on the chessboard

Þ Enumerate several independent sets at the same time

37

Geometric operator (1) n = 2 p Þ symmetry H

1 3 5 7 9 11 13 15 17 19 21 22 20 18 16 14 12 10 8 6 4 25 7 9 3 2 22 18 11 13 16 20 19 15 14 12 17 21 1 4 10 8 63 1 11 5 7 9 20 14 15 21 18 17 22 16 13 19 10 8 6 12 2 418 5 3 9 1 15 12 7 19 14 22 21 13 20 8 11 16 2 10 4 6 177 16 1 17 3 21 9 5 11 13 19 20 14 12 6 10 22 4 18 2 15 813 12 20 18 22 10 16 1 8 3 5 6 4 7 2 15 9 21 17 19 11 1415 21 19 1 14 8 6 12 4 17 10 9 18 3 11 5 7 13 2 20 22 1611 14 17 8 6 4 21 20 10 2 15 16 1 9 19 22 3 5 7 18 13 1210 8 4 14 21 19 2 18 16 6 12 11 5 15 17 1 20 22 13 3 7 920 6 12 2 10 17 15 22 7 4 14 13 3 8 21 16 18 9 1 11 5 192 10 15 4 8 6 11 13 21 20 17 18 19 22 14 12 5 7 3 16 9 14 13 6 12 19 18 22 2 9 8 16 15 7 10 1 21 17 20 11 5 14 38 2 10 11 4 16 5 19 18 22 13 14 21 17 20 6 15 3 12 9 1 716 9 8 21 5 14 17 3 20 1 11 12 2 19 4 18 13 6 22 7 10 156 4 7 20 17 13 1 21 12 15 9 10 16 11 22 2 14 18 19 8 3 522 15 2 13 18 20 7 9 5 11 4 3 12 6 10 8 19 17 14 1 16 2119 18 22 15 12 1 14 8 3 9 6 5 10 4 7 13 2 11 16 21 17 2017 20 21 16 11 3 10 6 14 7 2 1 8 13 5 9 4 12 15 22 19 189 22 14 6 20 12 4 16 1 18 8 7 17 2 15 3 11 19 5 13 21 1012 19 13 10 16 7 3 17 22 5 1 2 6 21 18 4 8 15 9 14 20 1121 17 16 19 13 5 8 10 2 12 3 4 11 1 9 7 6 14 20 15 18 2214 11 18 22 15 2 19 4 6 10 7 8 9 5 3 20 1 16 21 17 12 13

Search tree depth: n/2

(22) 22

38

Geometric operator (1) n = 2 p Þ symmetry H

1 3 5 7 9 11 13 15 17 19 21 22 20 18 16 14 12 10 8 6 4 25 7 9 3 2 22 18 11 13 16 20 19 15 14 12 17 21 1 4 10 8 63 1 11 5 7 9 20 14 15 21 18 17 22 16 13 19 10 8 6 12 2 418 5 3 9 1 15 12 7 19 14 22 21 13 20 8 11 16 2 10 4 6 177 16 1 17 3 21 9 5 11 13 19 20 14 12 6 10 22 4 18 2 15 813 12 20 18 22 10 16 1 8 3 5 6 4 7 2 15 9 21 17 19 11 1415 21 19 1 14 8 6 12 4 17 10 9 18 3 11 5 7 13 2 20 22 1611 14 17 8 6 4 21 20 10 2 15 16 1 9 19 22 3 5 7 18 13 1210 8 4 14 21 19 2 18 16 6 12 11 5 15 17 1 20 22 13 3 7 920 6 12 2 10 17 15 22 7 4 14 13 3 8 21 16 18 9 1 11 5 192 10 15 4 8 6 11 13 21 20 17 18 19 22 14 12 5 7 3 16 9 14 13 6 12 19 18 22 2 9 8 16 15 7 10 1 21 17 20 11 5 14 38 2 10 11 4 16 5 19 18 22 13 14 21 17 20 6 15 3 12 9 1 716 9 8 21 5 14 17 3 20 1 11 12 2 19 4 18 13 6 22 7 10 156 4 7 20 17 13 1 21 12 15 9 10 16 11 22 2 14 18 19 8 3 522 15 2 13 18 20 7 9 5 11 4 3 12 6 10 8 19 17 14 1 16 2119 18 22 15 12 1 14 8 3 9 6 5 10 4 7 13 2 11 16 21 17 2017 20 21 16 11 3 10 6 14 7 2 1 8 13 5 9 4 12 15 22 19 189 22 14 6 20 12 4 16 1 18 8 7 17 2 15 3 11 19 5 13 21 1012 19 13 10 16 7 3 17 22 5 1 2 6 21 18 4 8 15 9 14 20 1121 17 16 19 13 5 8 10 2 12 3 4 11 1 9 7 6 14 20 15 18 2214 11 18 22 15 2 19 4 6 10 7 8 9 5 3 20 1 16 21 17 12 13

Search tree depth: n/2

(22) 22

39

Geometric operator (1) n = 2 p Þ symmetry H

1 3 5 7 9 11 13 15 17 19 21 22 20 18 16 14 12 10 8 6 4 25 7 9 3 2 22 18 11 13 16 20 19 15 14 12 17 21 1 4 10 8 63 1 11 5 7 9 20 14 15 21 18 17 22 16 13 19 10 8 6 12 2 418 5 3 9 1 15 12 7 19 14 22 21 13 20 8 11 16 2 10 4 6 177 16 1 17 3 21 9 5 11 13 19 20 14 12 6 10 22 4 18 2 15 813 12 20 18 22 10 16 1 8 3 5 6 4 7 2 15 9 21 17 19 11 1415 21 19 1 14 8 6 12 4 17 10 9 18 3 11 5 7 13 2 20 22 1611 14 17 8 6 4 21 20 10 2 15 16 1 9 19 22 3 5 7 18 13 1210 8 4 14 21 19 2 18 16 6 12 11 5 15 17 1 20 22 13 3 7 920 6 12 2 10 17 15 22 7 4 14 13 3 8 21 16 18 9 1 11 5 192 10 15 4 8 6 11 13 21 20 17 18 19 22 14 12 5 7 3 16 9 14 13 6 12 19 18 22 2 9 8 16 15 7 10 1 21 17 20 11 5 14 38 2 10 11 4 16 5 19 18 22 13 14 21 17 20 6 15 3 12 9 1 716 9 8 21 5 14 17 3 20 1 11 12 2 19 4 18 13 6 22 7 10 156 4 7 20 17 13 1 21 12 15 9 10 16 11 22 2 14 18 19 8 3 522 15 2 13 18 20 7 9 5 11 4 3 12 6 10 8 19 17 14 1 16 2119 18 22 15 12 1 14 8 3 9 6 5 10 4 7 13 2 11 16 21 17 2017 20 21 16 11 3 10 6 14 7 2 1 8 13 5 9 4 12 15 22 19 189 22 14 6 20 12 4 16 1 18 8 7 17 2 15 3 11 19 5 13 21 1012 19 13 10 16 7 3 17 22 5 1 2 6 21 18 4 8 15 9 14 20 1121 17 16 19 13 5 8 10 2 12 3 4 11 1 9 7 6 14 20 15 18 2214 11 18 22 15 2 19 4 6 10 7 8 9 5 3 20 1 16 21 17 12 13

Search tree depth: n/2

(22) 22

40

Geometric operator (2) n = 3 p Þ central symmetry

Search tree depth: (n/2) - 1

(15) 15

1 10 2 3 8 4 11 12 7 14 13 15 9 6 513 3 1 6 2 5 15 10 8 9 12 11 14 7 415 7 5 11 1 9 14 2 6 4 3 13 12 10 89 12 14 7 6 8 1 4 5 2 11 10 15 3 13

11 6 3 4 9 15 12 13 10 8 14 1 5 2 712 14 8 13 3 11 2 5 4 1 6 7 10 15 94 2 12 5 10 6 13 7 3 11 15 9 8 14 15 11 4 1 7 14 10 15 9 13 8 2 3 12 62 13 7 10 15 12 4 8 14 5 9 6 11 1 3

10 15 9 8 5 2 3 6 1 12 4 14 7 13 118 1 6 2 13 7 9 14 11 15 10 3 4 5 12

14 4 15 9 12 1 6 3 2 7 5 8 13 11 107 9 11 14 4 3 5 1 13 10 2 12 6 8 153 8 13 12 11 10 7 9 15 6 1 5 2 4 146 5 10 15 14 13 8 11 12 3 7 4 1 9 2

41

Geometric operator (2) n = 3 p Þ central symmetry

Search tree depth: (n/2) - 1

(15) 15

1 10 2 3 8 4 11 12 7 14 13 15 9 6 513 3 1 6 2 5 15 10 8 9 12 11 14 7 415 7 5 11 1 9 14 2 6 4 3 13 12 10 89 12 14 7 6 8 1 4 5 2 11 10 15 3 13

11 6 3 4 9 15 12 13 10 8 14 1 5 2 712 14 8 13 3 11 2 5 4 1 6 7 10 15 94 2 12 5 10 6 13 7 3 11 15 9 8 14 15 11 4 1 7 14 10 15 9 13 8 2 3 12 62 13 7 10 15 12 4 8 14 5 9 6 11 1 3

10 15 9 8 5 2 3 6 1 12 4 14 7 13 118 1 6 2 13 7 9 14 11 15 10 3 4 5 12

14 4 15 9 12 1 6 3 2 7 5 8 13 11 107 9 11 14 4 3 5 1 13 10 2 12 6 8 153 8 13 12 11 10 7 9 15 6 1 5 2 4 146 5 10 15 14 13 8 11 12 3 7 4 1 9 2

42

Geometric operator (2) n = 3 p Þ central symmetry

Search tree depth: (n/2) - 1

(15) 15

1 10 2 3 8 4 11 12 7 14 13 15 9 6 513 3 1 6 2 5 15 10 8 9 12 11 14 7 415 7 5 11 1 9 14 2 6 4 3 13 12 10 89 12 14 7 6 8 1 4 5 2 11 10 15 3 13

11 6 3 4 9 15 12 13 10 8 14 1 5 2 712 14 8 13 3 11 2 5 4 1 6 7 10 15 94 2 12 5 10 6 13 7 3 11 15 9 8 14 15 11 4 1 7 14 10 15 9 13 8 2 3 12 62 13 7 10 15 12 4 8 14 5 9 6 11 1 3

10 15 9 8 5 2 3 6 1 12 4 14 7 13 118 1 6 2 13 7 9 14 11 15 10 3 4 5 12

14 4 15 9 12 1 6 3 2 7 5 8 13 11 107 9 11 14 4 3 5 1 13 10 2 12 6 8 153 8 13 12 11 10 7 9 15 6 1 5 2 4 146 5 10 15 14 13 8 11 12 3 7 4 1 9 2

43

Geometric operator (3) n = ( 4 p ) + 1 Þ /2 rotations:

R, R2 et R3

Search tree depth: (n/4) - 1

(21) 21

4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11

21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16

15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19

10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12

17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13

14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2

44

Geometric operator (3) n = ( 4 p ) + 1 Þ /2 rotations:

R, R2 et R3

Search tree depth: (n/4) - 1

(21) 21

4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11

21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16

15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19

10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12

17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13

14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2

45

Geometric operator (3) n = ( 4 p ) + 1 Þ /2 rotations:

R, R2 et R3

Search tree depth: (n/4) - 1

(21) 21

4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11

21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16

15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19

10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12

17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13

14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2

46

Geometric operator (3) n = ( 4 p ) + 1 Þ /2 rotations:

R, R2 et R3

Search tree depth: (n/4) - 1

(21) 21

4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11

21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16

15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19

10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12

17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13

14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2

47

Geometric operator (3) n = ( 4 p ) + 1 Þ /2 rotations:

R, R2 et R3

Search tree depth: (n/4) - 1

(21) 21

4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11

21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16

15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19

10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12

17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13

14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2

48

Geometric operator (3) n = ( 4 p ) + 1 Þ /2 rotations:

R, R2 et R3

Search tree depth: (n/4) - 1

(21) 21

4 8 12 16 13 6 15 20 2 10 19 17 9 5 11 14 1 21 7 18 319 10 4 8 17 11 2 12 5 6 1 15 18 14 21 20 16 3 13 9 78 14 17 9 4 13 16 15 1 12 5 2 21 19 6 10 18 7 20 3 11

21 4 8 18 10 9 6 2 19 11 14 1 5 16 20 13 3 17 12 7 152 13 19 4 8 14 5 10 18 1 6 11 15 17 12 21 7 9 3 20 16

15 17 11 14 21 3 7 1 4 8 9 18 6 20 19 2 13 12 16 10 512 21 7 17 9 20 13 11 3 18 10 4 19 2 16 6 8 5 15 1 146 15 20 13 18 17 3 5 12 7 2 21 16 8 10 4 9 1 14 11 19

10 19 21 6 16 7 20 13 9 5 15 14 12 11 2 3 17 18 4 8 118 16 3 2 12 19 1 21 15 14 20 13 8 6 17 7 4 10 11 5 920 2 6 15 7 10 11 3 16 17 21 19 14 1 9 12 5 13 8 4 1811 7 9 12 2 5 19 8 6 15 18 16 13 21 3 17 10 4 1 14 203 6 2 20 19 1 4 9 10 16 13 7 11 15 18 5 14 8 21 17 12

17 9 16 3 11 2 12 6 14 21 4 5 10 7 1 19 20 15 18 13 816 3 13 7 6 8 14 4 17 2 12 20 1 9 15 18 11 19 5 21 107 12 14 10 15 4 17 18 8 20 11 6 2 3 5 1 21 16 9 19 13

14 18 1 11 5 21 10 19 13 9 8 3 20 12 7 16 6 2 17 15 413 5 10 19 1 15 18 14 7 3 16 9 17 4 8 11 12 20 6 2 219 1 18 5 20 12 8 17 21 4 7 10 3 13 14 15 2 11 19 16 65 11 15 1 14 18 21 16 20 13 3 8 7 10 4 9 19 6 2 12 171 20 5 21 3 16 9 7 11 19 17 12 4 18 13 8 15 14 10 6 2

49

Geometric operator (4) n = ( 4 p ) Þ symmetries H & V

Search tree depth: (n/4)

(32) 32

a b c d e f g h i j k l m n o p q r s t u v w x y z & # @ $ * +c d a b g r e x t u h w k f q s n p & v j y l m i # o z * + @ $b e g c a t d q r k n f i y j u l w h x & s v o p @ m + $ z # *g a b e d o c t n h l p w v f x i & k j q u y s m $ r @ # * + zd m w a b e v p c r y u f i n z g s x & l h o $ q k # * + j t @e c q l m y z d a b s k o w x f & i j r v n * + @ g h t u p $ #m r e y c d p & l x a v b j z n s g w * k + i u f q @ $ h # o tz y d q o s u c j i b & a t # k v e m + f * x w $ l n r p @ h gq o n u t b & v # g j @ y c a i x + $ h d w z e k f * m l s r pu h x o q a @ g & v # s $ m b j w * t c n e k f z d + p r i y lp t l $ w z r # y s * i @ k + & f a v d x b n h e o g j c u m qw x z # * @ l $ + m r q h & s v k n f y p o t a c u d b e g i jf # u k x w + m @ * p $ s r g y h z o n c q b d t a j i v l e &x n + * $ k f j h o t z q @ l # e u d p g m r y w & v c b a s ik f o @ h n $ + * # i j u z t q p m g l w x e b a c s y d r & vn + * z k l w f p $ @ y e o i m t x r # h d c q & j u v g b a s$ k f h + m # * o n x g j p d l u @ q w z i s r b e t a y & v c+ * p x z $ n k f w d e t h u r o l y m # @ j & v s c g i q b a@ $ k f n + * e z p u h r x m w j t i o y l q g # b a s & v c d* w t + @ # k l x f o n c q y g z h p $ s r & i u v e d a m j b# @ h w f x m n k l q r z b c a + $ * g o p u v s t i & j y d eo u m n # h q w g c f d x + k b * v a i @ & $ z j p y e s t l rt z @ p r v x y b a w c n l & e # f u s $ j + * h i k o q d g mr p $ t u & b a w y e x g s v d @ k n z i # h j + * f l m c q oh g i r p c t s e d & b v u w + a j l k * f @ # n m $ q o x z yl q j g s i o b m @ v a & e h $ c y # f + k d t * r x n z w p uj s r m l g h i v & c + p # @ * b d e q a $ f k x y z u t o n wi l # v & j a o s q g t * d $ h y c @ b m z p n r + w f k e u xy v & j i p s u $ z m o # a * @ d b + e r t g c l n q x w f k h& j y s v u i r q + $ * d g e t m # z @ b c a p o x l k n h w fs i v & y q j @ u t z # + * r c $ o b a e g m l d w p h f k x nv & s i j * y z d e + m l $ p o r q c u t a # @ g h b w x n f k

50

Geometric operator (4) n = ( 4 p ) Þ symmetries H & V

Search tree depth: (n/4)

(32) 32

a b c d e f g h i j k l m n o p q r s t u v w x y z & # @ $ * +c d a b g r e x t u h w k f q s n p & v j y l m i # o z * + @ $b e g c a t d q r k n f i y j u l w h x & s v o p @ m + $ z # *g a b e d o c t n h l p w v f x i & k j q u y s m $ r @ # * + zd m w a b e v p c r y u f i n z g s x & l h o $ q k # * + j t @e c q l m y z d a b s k o w x f & i j r v n * + @ g h t u p $ #m r e y c d p & l x a v b j z n s g w * k + i u f q @ $ h # o tz y d q o s u c j i b & a t # k v e m + f * x w $ l n r p @ h gq o n u t b & v # g j @ y c a i x + $ h d w z e k f * m l s r pu h x o q a @ g & v # s $ m b j w * t c n e k f z d + p r i y lp t l $ w z r # y s * i @ k + & f a v d x b n h e o g j c u m qw x z # * @ l $ + m r q h & s v k n f y p o t a c u d b e g i jf # u k x w + m @ * p $ s r g y h z o n c q b d t a j i v l e &x n + * $ k f j h o t z q @ l # e u d p g m r y w & v c b a s ik f o @ h n $ + * # i j u z t q p m g l w x e b a c s y d r & vn + * z k l w f p $ @ y e o i m t x r # h d c q & j u v g b a s$ k f h + m # * o n x g j p d l u @ q w z i s r b e t a y & v c+ * p x z $ n k f w d e t h u r o l y m # @ j & v s c g i q b a@ $ k f n + * e z p u h r x m w j t i o y l q g # b a s & v c d* w t + @ # k l x f o n c q y g z h p $ s r & i u v e d a m j b# @ h w f x m n k l q r z b c a + $ * g o p u v s t i & j y d eo u m n # h q w g c f d x + k b * v a i @ & $ z j p y e s t l rt z @ p r v x y b a w c n l & e # f u s $ j + * h i k o q d g mr p $ t u & b a w y e x g s v d @ k n z i # h j + * f l m c q oh g i r p c t s e d & b v u w + a j l k * f @ # n m $ q o x z yl q j g s i o b m @ v a & e h $ c y # f + k d t * r x n z w p uj s r m l g h i v & c + p # @ * b d e q a $ f k x y z u t o n wi l # v & j a o s q g t * d $ h y c @ b m z p n r + w f k e u xy v & j i p s u $ z m o # a * @ d b + e r t g c l n q x w f k h& j y s v u i r q + $ * d g e t m # z @ b c a p o x l k n h w fs i v & y q j @ u t z # + * r c $ o b a e g m l d w p h f k x nv & s i j * y z d e + m l $ p o r q c u t a # @ g h b w x n f k

51

Geometric operator (4) n = ( 4 p ) Þ symmetries H & V

Search tree depth: (n/4)

(32) 32

a b c d e f g h i j k l m n o p q r s t u v w x y z & # @ $ * +c d a b g r e x t u h w k f q s n p & v j y l m i # o z * + @ $b e g c a t d q r k n f i y j u l w h x & s v o p @ m + $ z # *g a b e d o c t n h l p w v f x i & k j q u y s m $ r @ # * + zd m w a b e v p c r y u f i n z g s x & l h o $ q k # * + j t @e c q l m y z d a b s k o w x f & i j r v n * + @ g h t u p $ #m r e y c d p & l x a v b j z n s g w * k + i u f q @ $ h # o tz y d q o s u c j i b & a t # k v e m + f * x w $ l n r p @ h gq o n u t b & v # g j @ y c a i x + $ h d w z e k f * m l s r pu h x o q a @ g & v # s $ m b j w * t c n e k f z d + p r i y lp t l $ w z r # y s * i @ k + & f a v d x b n h e o g j c u m qw x z # * @ l $ + m r q h & s v k n f y p o t a c u d b e g i jf # u k x w + m @ * p $ s r g y h z o n c q b d t a j i v l e &x n + * $ k f j h o t z q @ l # e u d p g m r y w & v c b a s ik f o @ h n $ + * # i j u z t q p m g l w x e b a c s y d r & vn + * z k l w f p $ @ y e o i m t x r # h d c q & j u v g b a s$ k f h + m # * o n x g j p d l u @ q w z i s r b e t a y & v c+ * p x z $ n k f w d e t h u r o l y m # @ j & v s c g i q b a@ $ k f n + * e z p u h r x m w j t i o y l q g # b a s & v c d* w t + @ # k l x f o n c q y g z h p $ s r & i u v e d a m j b# @ h w f x m n k l q r z b c a + $ * g o p u v s t i & j y d eo u m n # h q w g c f d x + k b * v a i @ & $ z j p y e s t l rt z @ p r v x y b a w c n l & e # f u s $ j + * h i k o q d g mr p $ t u & b a w y e x g s v d @ k n z i # h j + * f l m c q oh g i r p c t s e d & b v u w + a j l k * f @ # n m $ q o x z yl q j g s i o b m @ v a & e h $ c y # f + k d t * r x n z w p uj s r m l g h i v & c + p # @ * b d e q a $ f k x y z u t o n wi l # v & j a o s q g t * d $ h y c @ b m z p n r + w f k e u xy v & j i p s u $ z m o # a * @ d b + e r t g c l n q x w f k h& j y s v u i r q + $ * d g e t m # z @ b c a p o x l k n h w fs i v & y q j @ u t z # + * r c $ o b a e g m l d w p h f k x nv & s i j * y z d e + m l $ p o r q c u t a # @ g h b w x n f k

52

Results synthesis

New results for the graphs counting more than ………………… 81 vertices (10) 11 1 sec. (20) 20 1 sec. (12) 12 1 sec. (21) 21 30844 sec. (14) 14 5 sec. (22) 22 233404 sec. (15) 15 4897 sec. (24) 24 10 sec. (16) 16 1 sec. (28) 28 1316 sec. (18) 18 2171 sec. (32) 32 73000 sec. … 1024 vertices

53

Results synthesis

Some improvements : fixing the first stable set (and its symmetric set) according to other certificates, branching heuristic , …

(26) 26 1,400,000 seconds

54

The 26 letters of the alphabet are enough for coloring the 26 X 26 chessboard

55

26 = ( 3 x 2 ) + …

56

26 = ( 5 x 4 ) + ( 3 x 2 )

57

(26) = 26 2216 sec. (better than 1,400,000)

58

30 = ( 6 x 4 ) + ( 3 x 2 )

“Too long” still more heuristic

Evaluate the nodes by counting the number of edges in the no colored sub graph

Partial branching : at each level of the search tree take only the best node among 10

59

X(30)=30cpu 2965 sec.

60

Results synthesis

Nous avons 13 contre exemples qui prouvent que n n’a pas besoin d’être premier avec 6 pour que (n) n

61

Outline

About the Queen Graph Coloring Problem Definition Intox/Conjecture ?

A Complete Algorithm Reformulation of the coloring problem Efficient filtering

A Geometric Based Heuristic Geometric Operators Results synthesis

Coloring Extension

62

Remark

An independent set with 12 vertices for n=12

63

… remains an independent set for n = 24 ...

Remark

64

Idea

… replacing 1 square that uses 1 color by 52 squares that use only 5 colors ...

65

Idea

… (5 12) = (60) = 60 ...

66

Complementary Diagonals

67

Polynomial formula for n no multiple of 2 or 3

r(i,j) ( 2.i + j ) modulo p

Þ complementary diagonals

68

Characteristic of Complementary Diagonals

69

Characteristic of Complementary Diagonals

70

Coloring Extension Formula

If (n) n and if p is prime with 2 and 3 then (np) np

given a coloring C(i,j) for the chessboard n n

r(i,j) ( 2i + j ) modulo p

R(i,j) = r(i,j) + p C(i/p,j/p)

(elementary algebra in the ring /p)

71

Coloring Extension Formula R(i,j) = r(i,j) + p.C(i/p,j/p)

If R(i,j) = R(i’,j)

Þ r(i,j) + p.C(i/p,j/p) = r(i,j) + p.C(i’/p,j/p)

Þ r(i,j)=r(i’,j) and C(i/p,j/p)= C(i’/p,j/p) because r < p

Þ i=i’ modulo p because 2 is prime with p ( recall : r(i,j) 2i+j )and

i/p=i’/p because C is a coloration and we are on the same column

then i = i’

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Coloring Extension Formula R(i,j) = r(i,j) + p.C(i/p,j/p)

Exercise

Prove the same kind of results for lines and diagonals

73

(60) (5 12) 5 12 60, (70) (5 14) 5 14 70, (75) (5 15) 5 15 75, (84) (7 12) 7 12 84,

… (2010) (67 30) 67 30 2010

…

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Conclusion

We know that (n) n for n 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 30 and 32

We know that there is an infinity of integers n that are multiples of 2 or 3 such that (n) n

Þ Is (n) n n 11 ?

But we don’t know, today, the value of (27)

75

Publications

Comptes Rendus Mathématiques de l’Académie des Sciences, Paris, Elsevier, Ser. I 342 (2006) p 157-160. “Coloration des graphes de reines”.

Journal of Heuristics (2004), 10 : p 407-413 : “New Results on Queen Graph Coloring Problem”.

ECAI'04, 16th European Conference on Artificial Intelligence, Valencia, Spain, August 22-27, 2004 : “Complete and Incomplete Algorithms for the Queen Graph Coloring Problem”.

Programmation en logique avec contraintes, JFPLC 2004, Hermes Science, ISBN 2-7462-0937-3, p 239-252 : “Algorithmes complet et incomplet pour la coloration des graphes de reines”.