2009/7/30, 南開大學, The Lit-Only Sigma Game on a Simple Graph

(1)

2009

The Lit-Only Sigma Game on a Simple Graph

Chih-wen weng

翁志文

Department of Applied Mathematics, National Chiao Tung University, Taiwan

2009

年

₇

月

₃₀

日

翁志文交通大學應用數學系年月日

(2)

Co-author: Hau-wen Huang (

黃皜文

₎

翁志文 (交通大學應用數學系) The Lit-Only Sigma Game on a Simple Graph 2009年7月 30日 2 / 39

(3)

2009

Lit-only sigma game

Let X = (S , E ) be a finite simple connected graph of order n. Every vertex of X can be assigned to either black state or white state to form a

configuration. Amoveon a configuration is to select one vertex s ∈ S having black state and then change those states of all neighbors of s.

Given two configurations, the goal is to decide if one can reach the other by a sequence of moves. This is the lit-only sigma gameon X .

(4)

Linear algebraic modeling

A configuration of the lit-only sigma game on the graph X = (X , E ) described in last page is naturally associated with a column vector u in the n-dimensional vector space F₂ⁿ over F₂ (n = |S |), where u_i = 1 iff the vertex i ∈ S is black. Each move is then naturally associated with an n × n matrix in GL_n(F₂) that acts on F₂ⁿby left multiplication.

(5)

2009年圖論與組合學國際學術會議暨第五屆海峽兩岸圖論與組合學術會議

Example A

b b

@@

@@ 1

2 3

4 5

6

r r

r

b b

@@

@@ 1

2 3

4 5 r 6

r r r







1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1











 1 1 0 1 0 0







=





 0 1 1 1 1 0







(6)

Example A

b b

@@

@@ 1

2 3

4 5

6

r r

r

b b

@@

@@ 1

2 3

4 5 r 6

r r r





0 1 0 0 0 0

0 1 1 0 0 0

0 0 0 1 0 0

0 1 0 0 1 0

0 0 0 0 0 1







 1 0 1 0 0





=



 1 1 1 1 0





(7)

Example A

b b

@@

@@ 1

2 3

4 5

6

r r

r

b b

@@

@@ 1

2 3

4 5 r 6

r r r







1 1 0 0 0 0

0 1 0 0 0 0

0 1 1 0 0 0

0 0 0 1 0 0

0 1 0 0 1 0

0 0 0 0 0 1











 1 1 0 1 0 0







=



 0 1 1 1 1 0





(8)

Example A

b b

@@

@@ 1

2 3

4 5

6

r r

r

b b

@@

@@ 1

2 3

4 5 r 6

r r r







1 1 0 0 0 0

0 1 0 0 0 0

0 1 1 0 0 0

0 0 0 1 0 0

0 1 0 0 1 0

0 0 0 0 0 1











 1 1 0 1 0 0







=





 0 1 1 1 1 0







(9)

Example B

b b

@@

@@ 1

2 3

4 5

6

r r

b b

@@

@@ 1

2 3

4 5

6

r r







1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1











 1 0 0 1 0 0







=





 1 0 0 1 0 0







(10)

Example B

b b

@@

@@ 1

2 3

4 5

6

r r

b b

@@

@@ 1

2 3

4 5

6

r r





0 1 0 0 0 0

0 1 1 0 0 0

0 0 0 1 0 0

0 1 0 0 1 0

0 0 0 0 0 1







 0 0 1 0 0





=



 0 0 1 0 0





(11)

Example B

b b

@@

@@ 1

2 3

4 5

6

r r

b b

@@

@@ 1

2 3

4 5

6

r r







1 1 0 0 0 0

0 1 0 0 0 0

0 1 1 0 0 0

0 0 0 1 0 0

0 1 0 0 1 0

0 0 0 0 0 1











 1 0 0 1 0 0







=



 1 0 0 1 0 0





(12)

Example B

b b

@@

@@ 1

2 3

4 5

6

r r

b b

@@

@@ 1

2 3

4 5

6

r r







1 1 0 0 0 0

0 1 0 0 0 0

0 1 1 0 0 0

0 0 0 1 0 0

0 1 0 0 1 0

0 0 0 0 0 1











 1 0 0 1 0 0







=





 1 0 0 1 0 0







(13)

2009

History I

This game implicitly appeared in M. Chuah (

蔡孟傑

) and C. Hu’s papers in 2004 when they studied the equivalence classes of Vogan diagrams. Gerald Jennhwa Chang (

張鎮華

) introduced this game to the Chinese

combinatorists by a talk in the title ”Graph Painting and Lie Algebra” in 2005 International and Third Cross-strait Conference on Graph Theory and

Combinatorics. (2005

年圖論與組合學國際學術會議暨第三屆海峽兩岸圖論

與組合學學術會議

) It was considered as a new game and the name of this game was not given when Chang’s talk was given.

(14)

History II

Xinmao Wang and Yaokun Wu recognized this game is a variety of anther game, called sigma game, which has been studied actively since 1980’s.

Even for the lit-only sigma game, M. Chuah and C. Hu were not the first two to study. It appears as early as in 2001 paper of H. Eriksson, K.

Eriksson, J. SjA¨ostrand.

(15)

Flipping groups and flipping classes

Definition

Let X = (X , E ) be a graph. For a vertex s ∈ S , we associate a matrix s ∈ Matn(F2), denoted by the bold type of s, as

s_uv =

1, if u = v , or v = s and uv ∈ E ; 0, else,

where u, v ∈ S .

Definition

Let W denote the subgroup of GL_n(F₂) generated by the set {s | s ∈ S }. W is referring to the flipping group of X .

Definition

The orbits of F₂ⁿ under W are called theflipping classes of X .

(16)

Flipping groups and flipping classes

Definition

s_uv =

where u, v ∈ S . Definition

Let W denote the subgroup of GL_n(F₂) generated by the set {s | s ∈ S }.

W is referring to the flipping group of X .

(17)

2009

Flipping groups and flipping classes

Definition

s_uv =

where u, v ∈ S . Definition

Let W denote the subgroup of GL_n(F₂) generated by the set {s | s ∈ S }.

W is referring to the flipping group of X . Definition

(18)

Dynkin diagram

Flipping classes of Dynkin Diagrams and extended Dynkin diagrams are determined by Meng-Kiat Chuah and Chu-Chin Hu in 2004, 2006 respectively.

An(n ≥ 1) _sb b b q q q b b b

n sn−1sn−2 s3 s2 s1

Dn(n ≥ 4) b

b b b q q q b b b

""

bb

sn−1

sn

sn−2sn−3 s3 s2 s1

E6 b b b b b

b

s5 s4 s3 s2 s1

s6

E7 b b b b b

b

s6 s5 s4 s3 s2 b

s7

s1

E8 b b b b b

b

s7 s6 s5 s4 s3 b b

s8

s2 s1

Figure: simply-laced Dynkin diagrams^翁志文 ⁽^{交通大學應用數學系}⁾ The Lit-Only Sigma Game on a Simple Graph 2009年7月 30日 10 / 39

(19)

2009

A graph with a long path

c c

c

c c c c c

sn−1 sn−2 sj_m sj₂ sj₁

sn

s2 s1

· · · · · · ·

C C C C C C C C CC

· · ·

Figure: The graph X = (S , E ).

(20)

Notations 1

Let S be a connected graph with n vertices s1, s2, . . . , sn that contains an induced path s₁, s₂, . . . , s_n−1 of n − 1 vertices, and s_n has neighbors s_j₁, s_j₂, . . . , s_j_m with 1 ≤ j₁ < j₂· · · < j_m≤ n − 1. Letes₁,es₂, . . . ,es_n denote the characteristic vectors of F₂ⁿ and let s1, s2, . . . , sn denote the flipping moves associated with s₁, s₂, . . . , s_n respectively.

Set

1 =es₁, i + 1 = s_is_i−1· · · s₁1 (1 ≤ i ≤ n − 1), n + 1 :=es_n. and consider the following three sets

Π = {1, 2, . . . , n},

Π₀ = {i ∈ Π | < i ,es_n>= 0}, Π₁ = Π − Π₀.

(21)

2009

Notations 2

By using the graph structure we can compute the following value

|Π₁| =

d^m₂e

X

k=1

j_2k− j_2k−1. Let

∆ :=

Π, if |Π₁| is odd;

Π ∪ {n + 1} − {n}, if |Π1| is even

be the simple basis of F₂ⁿ as shown in the beginning of Section ??. For a vector u ∈ F₂ⁿ let sw (u) denote the simple weight of u, i.e. the number nonzero terms in writing u as a linear combination of elements in ∆. Let U be the subspace spanned by the vectors in Π. For V ⊆ F₂ⁿ and

T ⊆ {0, 1, . . . , n},

V_T := {u ∈ V | sw (u) ∈ T },

and for shortness V_t₁_,t₂_,...,t_i := V_{t₁_,t₂_,...,t_i_}. Let odd be the subset of {1, 2, . . . , n} consisting of odd integers.

(22)

Notations 3

Set

Ai = {j ∈ [n] | j ≡ i , n + |Π₁| − i (mod 4)},

B_i = {j ∈ [n − 1] | j ≡ i , i + |Π₁| − 2, n − i , n − i + |Π₁| − 2 (mod 4)}, C_i = {j ∈ [n] | j ≡ i , i + |Π₁|, n + 2 − i , n + 2 − i + |Π₁| (mod 4)}.

Let P denote the set of orbits of the flipping puzzle on S . Then the set P and its cardinality |P| are given in the following table according to the different cases of the pair (|Π1|, n) in the first two columns.

(23)

2009

Flipping classes of a graph with a long path

|Π₁| n nontrivial O ∈ P

(might be repeated) |P|

3 ≤ |Π₁| ≤ n − 3,

|Π1| is odd even UAj 3

3 ≤ |Π1| ≤ n − 3,

|Π1| is odd odd U_A_j 4

4 ≤ |Π1| ≤ n − 3,

|Π1| is even even U_B_j, U_C_j 6

(24)

4 ≤ |Π₁| ≤ n − 3,

|Π₁| is even odd U_B_j, U_C_j 4

|Π₁| = 1 U_t,n+1−t d(n + 2)/2e

|Π₁| = 2 even U_{i ,n−i}, U_C₁, U_C₂ (n + 6)/2

|Π₁| = 2 odd Ui ,n−i, UC1, UC2 (n + 3)/2

|Π1| = n − 2,

|Π1| is odd odd U_odd, U_2i (n + 3)/2

|Π1| = n − 2,

|Π1| is even even Uodd, U2h,n−2h,

U_odd, U_{2g ,n+2−2g} (n + 6)/2

|Π1| = n − 1,

|Π1| is odd even U2t−1,2t (n + 2)/2

|Π1| = n − 1,

|Π1| is even odd U2h−1,2h,n−2h,,n+1−2h, U2g −1,2gn+2−2g ,n+3−2g

(n + 3)/2

(25)

2009

Flipping classes of line graphs

Yaokun Wu, Lit-only sigma game on a line graph, European Journal of Combinatorics 30(2009), 84-95.

(26)

Problems

Determine the flipping classes of X when X is a chessboard.

e e e e

(27)

Maximum-orbit-weight

For u ∈ F₂ⁿ, let w (u) denotes the Hamming weight of u, and for an flipping class O of X , w (O) := min{w (u) | u ∈ O} is called theweight of the flipping class O. The number

M(X ) := max{w (O) | O ∈ P}

is called the maximum-orbit-weightof the graph S .

1 M(X ) = 1 if X is one of simply-laced Dynkin diagrams [Chuah and Hu, 2004] (Borel-de Siebenthal Theorem).

2 [X. Wang, Y. Wu, 2007] and [H. Wu, G. J. Chang, 2006] independently show M(X ) ≤ d`/2e if X is a tree with ` leaves.

3 [Y. Wu, 2009] discovers that if X is the line graph of a simple graph Γ, then there is a close connection between M(X ) and the edge isoperimetric number of Γ.

4 When X has a long path, we give a necessary and sufficient condition for M(X ) = 1.

(28)

Maximum-orbit-weight

M(X ) := max{w (O) | O ∈ P}

independently show M(X ) ≤ d`/2e if X is a tree with ` leaves.

(29)

Maximum-orbit-weight

M(X ) := max{w (O) | O ∈ P}

2 [X. Wang, Y. Wu, 2007] and [H. Wu, G. J. Chang, 2006]

(30)

Maximum-orbit-weight

M(X ) := max{w (O) | O ∈ P}

for M(X ) = 1.

(31)

2009

Maximum-orbit-weight

M(X ) := max{w (O) | O ∈ P}

(32)

Six alternative moves 2, 5, 2, 5, 2, 5 of the edge 25

b b

r b

r r

@@

1 2

3

5 4

6 r b

r r

b r

@@

1 2

3

5 4 6

r r

b r

b b

@@

1 2

3

5 4

6 r r

b r

b b

@@

1 2

3

5 4 6

r b

r r

b r

@@

1 2

3

5 4

6 b b

r b

r r

@@

1 2

3

5 4 6

(33)

2009

Coxeter group associated a graph

Let X = (S , E ) denote a simple connected graph with vertex set {s₁, s₂, . . . , s_n}.

Definition

The Coxeter group W := W (X ) of a simple connected graph X = (S , R) is the group with the set S = {s_i | 1 ≤ i ≤ n} of generators subject only to relations

s_i² = 1,

(s_is_j)³ = 1, if ij ∈ E , (sisj)² = 1, if ij 6∈ E .

(34)

Relation between Coxeter group and flipping group

1 There is a homomorphism for the Coxeter group W of X onto the flipping group W sending generator s to the move s.

3 If |W | < ∞ then W /Z (W ) ∼= W, where Z (W ) is the center of the Coxter group W of X ; moreover, |Z (W )| ≤ 2.

4 Among all n-vertex graphs containing an induced (n − 1)-vertex path, there are at most n − 1 flipping groups up to isomorphism.

5 If X is the line graph of a graph with m edges and n vertices, then the flipping group W of X isomorphic to (Z/2Z)(n−1)(m−n+1)

o Sn if n is odd; (Z/2Z)(n−2)(m−n+1)

o Sn if n is even.

(35)

Relation between Coxeter group and flipping group

2 The center Z (W) of the flipping group W of X is trivial.

o Sn if n is even.

(36)

Relation between Coxeter group and flipping group

there are at most n − 1 flipping groups up to isomorphism.

o Sn if n is even.

(37)

Relation between Coxeter group and flipping group

o Sn if n is even.

(38)

Relation between Coxeter group and flipping group

o Sn if n is even.

(39)

2009

Reeder’s game

A configuration is an assignment of one of two color, black or white, to each vertex of X . A move applied on a configuration is to select a vertex v having an odd number of black neighbors and change the color of v .

(40)

Reeder’s game

Let X be

e e

@

@@

@

@@ s₁

s2

s₃

s₄ s5

s₆

u u

uu u

x = s1+ s2+ s4+ s5

ρ₁(x ) =?

ρ₁(x ) = ρ₁(s₁+ s₂+ s₄+ s₅) = s₂+ s₄+ s₅ ρ5(ρ1(x )) =?

ρ5(ρ1(x )) = s2+ s4+ s5

(41)

2009

Reeder’s game

Let X be

e e

@

@@

@

@@ s₁

s2

s₃

s₄ s5

s₆

u u

uu u

x = s1+ s2+ s4+ s5

ρ₁(x ) =?

ρ₁(x ) = ρ₁(s₁+ s₂+ s₄+ s₅) = s₂+ s₄+ s₅ ρ₅(ρ₁(x )) =?

ρ5(ρ1(x )) = s2+ s4+ s5

(42)

Reeder’s game

Let X be

e e

@

@@

@

@@ s₁

s2

s₃

s₄ s5

s₆

u u

uu u

x = s1+ s2+ s4+ s5

ρ₁(x ) =?

ρ₁(x ) = ρ₁(s₁+ s₂+ s₄+ s₅) = s₂+ s₄+ s₅ ρ₅(ρ₁(x )) =?

ρ5(ρ1(x )) = s2+ s4+ s5

(43)

2009

Reeder’s game

Let X be

e e

@

@@

@

@@ s₁

s2

s₃

s₄ s5

s₆

u u

uu u

x = s1+ s2+ s4+ s5

ρ₁(x ) =?

ρ₁(x ) = ρ₁(s₁+ s₂+ s₄+ s₅) = s₂+ s₄+ s₅ ρ₅(ρ₁(x )) =?

ρ5(ρ1(x )) = s2+ s4+ s5

(44)

Reeder’s game

Let X be

e e

@

@@

@

@@ s₁

s2

s₃

s₄ s5

s₆

u u

uu u

x = s1+ s2+ s4+ s5

ρ1(x ) =?

ρ₁(x ) = ρ₁(s₁+ s₂+ s₄+ s₅) = s₂+ s₄+ s₅ ρ₅(ρ₁(x )) =?

ρ5(ρ1(x )) = s2+ s4+ s5

(45)

2009

Reeder’s game

Let X be

e e

@

@@

@

@@ s₁

s2

s₃

s₄ s5

s₆ u

uu u

x = s1+ s2+ s4+ s5

ρ1(x ) =?

ρ1(x ) = ρ1(s1+ s2+ s4+ s5) = s2+ s4+ s5

ρ₅(ρ₁(x )) =?

ρ5(ρ1(x )) = s2+ s4+ s5

(46)

Reeder’s game

Let X be

e e

@

@@

@

@@ s₁

s2

s₃

s₄ s5

s₆ u

uu u

x = s1+ s2+ s4+ s5

ρ1(x ) =?

ρ1(x ) = ρ1(s1+ s2+ s4+ s5) = s2+ s4+ s5

ρ5(ρ1(x )) =?

ρ5(ρ1(x )) = s2+ s4+ s5

(47)

2009

Reeder’s game

Let X be

e e

@

@@

@

@@ s₁

s2

s₃

s₄ s5

s₆ u

uu u

x = s1+ s2+ s4+ s5

ρ1(x ) =?

ρ1(x ) = ρ1(s1+ s2+ s4+ s5) = s2+ s4+ s5

ρ5(ρ1(x )) =?

ρ5(ρ1(x )) = s2+ s4+ s5

(48)

Reeder’s game

Let X be

e e

@

@@

@

@@ s1

s₂ s3

s4

s₅ s6

u

uu u

x = s₁+ s₂+ s₄+ s₅ ρ₁(x ) =?

ρ₁(x ) = ρ₁(s₁+ s₂+ s₄+ s₅) = s₂+ s₄+ s₅ ρ₅(ρ₁(x )) =?

ρ₅(ρ₁(x )) = s₂+ s₄+ s₅

(49)

2009

Duality between Reeder’s game and lit-only σ-game

e e

@

@@

@

@@ 1

2

3

4 5 u 6

u u u





 0 1 1 1 1 0







t

=





 0 1 1 1 1 0







t





1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1











 0 1 1 1 1 0







=





 1 1 0 1 0 0







(50)

Duality

The orbits of Reeder’s game are called Reeder’s classes. A graph X is nonsingular if the determinant det(A) = 1 in F₂, where A is the adjacency matrix of X .

Lemma

Suppose that X is a nonsingular graph. Then there exists a bijection between flipping classes and Reeder’s classes.

(51)

2009

Reeder’s Theorem

Theorem (2005, M. Reeder)

Suppose that X is a tree with a perfect matching, not a path. Then there are exactly three Reeder’s classes on X .

(52)

Orbits distinguishing

Theorem (2009, J. Goldwasser, X. Wang, Y. Wu)

Suppose that X is a nonsingular graph of n vertices. Let u ∈ F₂ⁿ be a configuration with u_i = 0 for some i . Let A_i denote the i -th column of the adjacency matrix A. Then u and u + Ai are in two different flipping classes.

(53)

Applications

By the dual connection between Reeder’s game and lit-only σ-game, and using J. Goldwasser, X. Wang, Y. Wu’s Theorem to distinguish flipping classes, Hau-wen Huang can show

Corollary

Suppose that X is a tree with a perfect matching(equivalent to X nonsingular), not a path. Then there are exactly three flipping classes. Furthermore the maximum-orbit-weight M(X ) = 1.

Problem: Find an algorithm to do this.

(54)

Applications

Corollary

Suppose that X is a tree with a perfect matching(equivalent to X nonsingular), not a path. Then there are exactly three flipping classes.

Furthermore the maximum-orbit-weight M(X ) = 1.

(55)

Applications

Corollary

(56)

Applications

Corollary

(57)

Generalization

Suppose that X is a nonsingular graph, not a line graph.

Then there are exactly three flipping classes of X .

Moreover, M(X ) ≤ 2 (Hau-wen Huang, preprint).

The dual version of the above result does not appear in Reeder’s 2005 paper.

Problem. Characterize the case M(X ) = 1 when X is nonsingular.

(58)

Generalization

Suppose that X is a nonsingular graph, not a line graph. Then there are exactly three flipping classes of X .

(59)

Generalization

(60)

Generalization

(61)

2009

Generalization

(62)

Revisit the move on Reeder’s game

Let u ∈ F₂ⁿbe a configuration of Reeder’s game on X = (S , E ). Let s be the n × n move matrix associate with the vertex s ∈ S . We also use s to denote the characteristic vector of s ∈ S . Let fs(u) denote the new

configuration from u by applying the move s in Reeder’s game on X . Then fs(u)^t = u^ts

= u^t+ (u^tAs) s^t

= u^t+ < u, s > s^t, where < u, s >:= u^tAs is the inner product.

s

(63)

2009

Revisit the move on Reeder’s game

Let u ∈ F₂ⁿbe a configuration of Reeder’s game on X = (S , E ). Let s be the n × n move matrix associate with the vertex s ∈ S . We also use s to denote the characteristic vector of s ∈ S . Let fs(u) denote the new

configuration from u by applying the move s in Reeder’s game on X . Then fs(u)^t = u^ts

= u^t+ (u^tAs) s^t

= u^t+ < u, s > s^t, where < u, s >:= u^tAs is the inner product.

The above function f_s is called atransvectionin the literature.

(64)

Search for transvection

(65)

2009

83 matches

(66)

Since 1892

(67)

2009

2009/7/30, 南開大學, The Lit-Only Sigma Game on a Simple Graph

The Lit-Only Sigma Game on a Simple Graph

翁志文

年

月

日

黃皜文

Lit-only sigma game

Linear algebraic modeling

Example A

Example A

Example A

Example A

Example B

Example B

Example B

Example B

History I

蔡孟傑

張鎮華

年圖論與組合學國際學術會議暨第三屆海峽兩岸圖論

與組合學學術會議

History II

Flipping groups and flipping classes

Flipping groups and flipping classes

Flipping groups and flipping classes

Dynkin diagram

A graph with a long path

Notations 1

Notations 2

Notations 3

Flipping classes of a graph with a long path

Flipping classes of line graphs

Problems

Maximum-orbit-weight

Maximum-orbit-weight

Maximum-orbit-weight

Maximum-orbit-weight

Maximum-orbit-weight

Six alternative moves 2, 5, 2, 5, 2, 5 of the edge 25

Coxeter group associated a graph

Relation between Coxeter group and flipping group

Relation between Coxeter group and flipping group

Relation between Coxeter group and flipping group

Relation between Coxeter group and flipping group

Relation between Coxeter group and flipping group

Reeder’s game

Reeder’s game

Reeder’s game

Reeder’s game

Reeder’s game

Reeder’s game

Reeder’s game

Reeder’s game

Reeder’s game

Reeder’s game

Duality between Reeder’s game and lit-only σ-game

Duality

Reeder’s Theorem

Orbits distinguishing

Applications

Applications

Applications

Applications

Generalization

Generalization

Generalization

Generalization

Generalization

Revisit the move on Reeder’s game

Revisit the move on Reeder’s game

Thank you for your attention.