Colourings of (0 , m )-graphs and the switching operation Gary - - PowerPoint PPT Presentation

Colourings of (0, m)-graphs and the switching operation

Gary MacGillivray University of Victoria Joint work with Chris Duffy, Ben Tremblay and J. Maria Warren GT2015, Nyborg, DK, August 26, 2015

Prelude

◮ Harary (1953) used graphs with 2 edge colours to express

preferences.

◮ Albeson and Rosenborg (1958) introduced a operation that

reverses all preferences at a vertex. Vendors Consumers

Prelude

◮ Harary (1953) used graphs with 2 edge colours to express

preferences.

◮ Albeson and Rosenborg (1958) introduced a operation that

reverses all preferences at a vertex. Vendors Consumers

Prelude

◮ Harary (1953) used graphs with 2 edge colours to express

preferences.

◮ Albeson and Rosenborg (1958) introduced a operation that

reverses all preferences at a vertex. Vendors Consumers

Prelude

◮ Harary (1953) used graphs with 2 edge colours to express

preferences.

◮ Albeson and Rosenborg (1958) introduced a operation that

reverses all preferences at a vertex. Vendors Consumers What collections of preferences are possible − if only consumers can switch? − if both consumers and vendors can switch?

Edge-coloured graphs

An m-edge-coloured graph is obtained from a simple graph by assigning each edge one of m available colours.

Edge-coloured graphs

An m-edge-coloured graph is obtained from a simple graph by assigning each edge one of m available colours. An m-edge coloured graph, G = (V , E0, E1, . . . , Em−1), consists of a vertex set, V , and m disjoint edge sets, E0, E1, . . . , Em−1.

Homomorphisms and colourings

A homomorphism G → H is a mapping of the vertices of G to the vertices of H that preserves edges and edge colours.

1 2 3 1 4 1 2 3

G

1 2 3 4

H →

Homomorphisms and colourings

A homomorphism G → H is a mapping of the vertices of G to the vertices of H that preserves edges and edge colours.

1 2 3 1 4 1 2 3

G

1 2 3 4

H → A k-colouring of G is a homomorphism to an m-edge-coloured (complete) graph on k vertices.

Homomorphisms and colourings

A homomorphism G → H is a mapping of the vertices of G to the vertices of H that preserves edges and edge colours.

1 2 3 1 4 1 2 3

G

1 2 3 4

H → A k-colouring of G is a homomorphism to an m-edge-coloured (complete) graph on k vertices. The vertex colours respect the edge types.

Chromatic number

χe(G): minimum k such that G has a k-colouring. (Why χe?)

1 2 3 1 ? 1 2 3

G χe(G) ≤ 4. Equality?

Chromatic number

χe(G): minimum k such that G has a k-colouring. (Why χe?)

1 2 3 1 ? 1 2 3

G χe(G) ≤ 4. Equality? Yes.

Weak duality

Theorem (Kostochka, Sopena, Zhu, 1997)

Let G be an orientation of a graph with maximum degree ∆. Then χo(G) ≤ 2∆22∆.

Weak duality

Theorem (Kostochka, Sopena, Zhu, 1997)

Let G be an orientation of a graph with maximum degree ∆. Then χo(G) ≤ 2∆22∆. The statement also holds, with the same proof method, if − orientation is replaced by 2-edge-colouring, and − χo is replaced by χe.

Weak duality

Theorem (Kostochka, Sopena, Zhu, 1997)

Let G be an orientation of a graph with maximum degree ∆. Then χo(G) ≤ 2∆22∆. The statement also holds, with the same proof method, if − orientation is replaced by 2-edge-colouring, and − χo is replaced by χe. This sort phenomenon happens frequently.

Weak duality

Theorem (Kostochka, Sopena, Zhu, 1997)

Let G be an orientation of a graph with maximum degree ∆. Then χo(G) ≤ 2∆22∆. The statement also holds, with the same proof method, if − orientation is replaced by 2-edge-colouring, and − χo is replaced by χe. This sort phenomenon happens frequently. Not always: − Any orientation of P5 has χo ≤ 3; − An alternating 2-edge-colouring of P5 has χe = 4.

3 2 1 1 ?

Mixed graphs

Neˇ setˇ ril and Raspaud (2000) define (n, m)-mixed graphs:

◮ n disjoint arc sets and m disjoint edge sets; and ◮ any two vertices are joined by at most one edge or arc.

Results that hold for these hold for (1, 0)-mixed graphs (oriented graphs) and (0, 2)-mixed graphs (2-edge-coloured graphs). (0, m)-mixed graphs are m-edge-coloured graphs.

Mixed graphs

Neˇ setˇ ril and Raspaud (2000) define (n, m)-mixed graphs:

◮ n disjoint arc sets and m disjoint edge sets; and ◮ any two vertices are joined by at most one edge or arc.

Results that hold for these hold for (1, 0)-mixed graphs (oriented graphs) and (0, 2)-mixed graphs (2-edge-coloured graphs). (0, m)-mixed graphs are m-edge-coloured graphs.

Theorem (Kostochka, Sopena, Zhu, 1997)

Let G be an orientation of a graph with maximum degree ∆. Then χo(G) ≤ 2∆22∆.

Mixed graphs

Neˇ setˇ ril and Raspaud (2000) define (n, m)-mixed graphs:

◮ n disjoint arc sets and m disjoint edge sets; and ◮ any two vertices are joined by at most one edge or arc.

Results that hold for these hold for (1, 0)-mixed graphs (oriented graphs) and (0, 2)-mixed graphs (2-edge-coloured graphs). (0, m)-mixed graphs are m-edge-coloured graphs.

Theorem (2015)

Let G be an m-edge-colouring of a graph with max. degree ∆. Then χe(G) ≤ m∆2m∆.

Switching

◮ Switching in 2-edge-coloured graphs is well-studied (signed

graphs).

◮ Seidel switching corresponds to switching in 2-edge-coloured

complete graphs.

◮ Switching (arc reversal) in oriented graphs is fairly well

studied (pushing vertices).

◮ Switching in m-edge coloured graphs has been considered. A

switch corresponds a cyclic permutation of the edge colours at a vertex.

◮ We want to generalize and allow richer collections of

permutations.

Switching

Let G be an m-edge-coloured graph, Γ ≤ Sm, and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours

• f the edges incident with x according to γ.

Switching

Let G be an m-edge-coloured graph, Γ ≤ Sm, and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours

• f the edges incident with x according to γ.

Suppose π = (blue black red), σ = (red black blue) ∈ Γ.

Switching

Let G be an m-edge-coloured graph, Γ ≤ Sm, and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours

• f the edges incident with x according to γ.

π Suppose π = (blue black red), σ = (red black blue) ∈ Γ.

Switching

Let G be an m-edge-coloured graph, Γ ≤ Sm, and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours

• f the edges incident with x according to γ.

π Suppose π = (blue black red), σ = (red black blue) ∈ Γ.

Switching

Let G be an m-edge-coloured graph, Γ ≤ Sm, and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours

• f the edges incident with x according to γ.

σ Suppose π = (blue black red), σ = (red black blue) ∈ Γ.

Switching

Let G be an m-edge-coloured graph, Γ ≤ Sm, and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours

• f the edges incident with x according to γ.

σ Suppose π = (blue black red), σ = (red black blue) ∈ Γ.

Switching

Let G be an m-edge-coloured graph, Γ ≤ Sm, and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours

• f the edges incident with x according to γ.

σ Suppose π = (blue black red), σ = (red black blue) ∈ Γ. G and H are switch equivalent with respect to Γ if some sequence

• f switches transforms G so that it becomes isomorphic to H.

Some groups might be less interesting

Suppose Γ ≤ Sm has Property Tj: − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k. Using Property Tj we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S3 has this property for all j. Say j is red.

x y

Some groups might be less interesting

Suppose Γ ≤ Sm has Property Tj: − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k. Using Property Tj we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S3 has this property for all j. Say j is red.

x y

α = (black red)(blue), β = (red blue black)

Some groups might be less interesting

Suppose Γ ≤ Sm has Property Tj: − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k. Using Property Tj we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S3 has this property for all j. Say j is red.

x y

α α = (black red)(blue), β = (red blue black)

Some groups might be less interesting

Suppose Γ ≤ Sm has Property Tj: − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k. Using Property Tj we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S3 has this property for all j. Say j is red.

x y

α α = (black red)(blue), β = (red blue black)

Some groups might be less interesting

Suppose Γ ≤ Sm has Property Tj: − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k. Using Property Tj we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S3 has this property for all j. Say j is red.

x y

α β α = (black red)(blue), β = (red blue black)

Some groups might be less interesting

Suppose Γ ≤ Sm has Property Tj: − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k. Using Property Tj we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S3 has this property for all j. Say j is red.

x y

α β α = (black red)(blue), β = (red blue black)

Some groups might be less interesting

Suppose Γ ≤ Sm has Property Tj: − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k. Using Property Tj we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S3 has this property for all j. Say j is red.

x y

α β α−1 α = (black red)(blue), β = (red blue black)

Some groups might be less interesting

Suppose Γ ≤ Sm has Property Tj: − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k. Using Property Tj we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S3 has this property for all j. Say j is red.

x y

α β α−1 α = (black red)(blue), β = (red blue black)

Some groups might be less interesting

Suppose Γ ≤ Sm has Property Tj: − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k. Using Property Tj we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S3 has this property for all j. Say j is red.

x y

α β α−1 β−1 α = (black red)(blue), β = (red blue black)

Some groups might be less interesting

Suppose Γ ≤ Sm has Property Tj: − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k. Using Property Tj we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S3 has this property for all j. Say j is red.

x y

α β α−1 β−1 α = (black red)(blue), β = (red blue black)

One equivalence class

Theorem (2015)

If Γ ≤ Sm has Property Tj for all j, then any two m-edge-coloured graphs with isomorphic underlying graphs are switch equivalent w.r.t. Γ.

Property Tj: Γ acts transitively, and for all i there exists α ∈ Γ that sends i to j and fixes some k.

One equivalence class

Theorem (2015)

If Γ ≤ Sm has Property Tj for all j, then any two m-edge-coloured graphs with isomorphic underlying graphs are switch equivalent w.r.t. Γ. Groups with Property Tj for all j include: − the symmetric group Sm, m ≥ 3, − the alternating group Am, m ≥ 4, and − the dihedral group D2t+1, t ≥ 1 (symmetries of a (2t + 1)-gon), but not D2t, t ≥ 1.

Property Tj: Γ acts transitively, and for all i there exists α ∈ Γ that sends i to j and fixes some k.

Switching with Dm, m even

Note that D2 = S2 = Z2.

Theorem (Brewster and Graves, 2009)

The m-edge-coloured graphs G and H are switch equivalent with respect to Zm if and only if PZm(G) ∼ = PZm(H). G e (r b) PZ2(G)

Switch equivalence w.r.t. Dm, m even

Let E = {0, 2, . . . , m − 2}, and O = {1, 2, . . . , m − 1}. Then E, O is a block system for the action of Dm.

Switch equivalence w.r.t. Dm, m even

Let E = {0, 2, . . . , m − 2}, and O = {1, 2, . . . , m − 1}. Then E, O is a block system for the action of Dm. Let G be an m-edge-coloured graph. Obtain G2 by recolouring each edge with a colour in E by 0, and each edge with a colour in O by 1.

Switch equivalence w.r.t. Dm, m even

Let E = {0, 2, . . . , m − 2}, and O = {1, 2, . . . , m − 1}. Then E, O is a block system for the action of Dm. Let G be an m-edge-coloured graph. Obtain G2 by recolouring each edge with a colour in E by 0, and each edge with a colour in O by 1.

Theorem (2015)

For even m, the m-edge-coloured graphs G and H are switch equivalent with respect to Dm if and only if G2 and H2 are switch equivalent with respect to Z2.

More about switch equivalence w.r.t. Dm, m even

Theorem (Zaslavski, 1982)

The 2-edge-coloured graphs G and H are switch equivalent with respect to Z2 if and only if some G ′ ∼ = G has the same collection

• f cycles with an odd number of red edges as H.

More about switch equivalence w.r.t. Dm, m even

Theorem (Zaslavski, 1982)

The 2-edge-coloured graphs G and H are switch equivalent with respect to Z2 if and only if some G ′ ∼ = G has the same collection

• f cycles with an odd number of red edges as H.

Are these switch equivalent with respect to Z2? G H

More about switch equivalence w.r.t. Dm, m even

Theorem (Zaslavski, 1982)

The 2-edge-coloured graphs G and H are switch equivalent with respect to Z2 if and only if some G ′ ∼ = G has the same collection

• f cycles with an odd number of red edges as H.

Corollary

For even m, G and H are switch equivalent with respect to Dm if and only if some G ′ ∼ = G has the same collection of cycles with an

• dd number of elements in E as H.

Switch equivalence w.r.t. Abelian groups

Theorem (Brewster and Graves, 2009)

The m-edge-coloured graphs G and H are switch equivalent with respect to Zm if and only if PZm(G) ∼ = PZm(H).

Theorem (2015)

Let Γ ≤ Sm be Abelian. For m-edge-coloured graphs G and H, there exist m-edge coloured graphs PΓ(G) and PΓ(H) such that G and H are switch equivalent with respect to Γ if and only if PΓ(G) ∼ = PΓ(H).

PZ3(G)

a b c d

G

PZ3(G)

a b c d

G

e (0 1 2) (0 2 1)

PZ3(G)

PZ3(G)

a b c d

G

e (0 1 2) (0 2 1)

PZ3(G) With Abelian groups it is enough to have one switch at each vertex (perhaps w.r.t. e).

Γ-switchable colouring

Let G be an m-edge-coloured graph. Γ-switchable chromatic number, χΓ(G): minimum k s.t. some G ′ switch equivalent to G has a k-colouring.

◮ If Γ is trivial, then χΓ(G) = χe(G).

(Ah-ha?)

◮ If Γ ≤ Sm has Property Tj for some j, then χΓ(G) is the

chromatic number of the underlying graph. ∴ for Sm, m ≥ 3, Am, m ≥ 4 and Dm, odd m ≥ 3.

◮ For m even, χDm(G) = χZ2(G2).

Γ-switchable homomorphisms

G →Γ H: there is a sequence of switches w.r.t. Γ that transform G so that it has a homomorphism to H.

Γ-switchable homomorphisms

G →Γ H: there is a sequence of switches w.r.t. Γ that transform G so that it has a homomorphism to H.

Theorem (Brewster and Graves, 2009)

Let G and H be m-edge-coloured graphs. Then G →Zm H if and

• nly if G → PZm(H).

Γ-switchable homomorphisms

G →Γ H: there is a sequence of switches w.r.t. Γ that transform G so that it has a homomorphism to H.

Theorem (2015)

Let G and H be m-edge-coloured graphs. For any Abelian group Γ, G →Γ H if and only if G → PΓ(H).

Corollary

If Γ ≤ Sm is Abelian, then χΓ(G) ≤ χe(G) ≤ |Γ|χΓ(G).

Complexity

Theorem

Suppose Γ ≤ Sm is Abelian or D2t, or has Property Tj for some j. Then, Γ-switchable 2-colouring is Polynomial, and for all k ≥ 3, Γ-switchable k-colouring is NP-complete. PZ3(K2) Proof example. Γ = Z3. acting on {red, black, blue}. Any two monochromatic K2s are switch equivalent w.r.t. Z3. Thus, G is Γ-switchably 2-colourable if and only if there is a homomorphism G → PZ3(K2). This can be decided in Polynomial time.