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 , E 0 , E 1 , . . . , E m − 1 ), consists of a vertex set, V , and m disjoint edge sets, E 0 , E 1 , . . . , E m − 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. 4 1 1 2 2 3 3 3 1 4 2 1 → G 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. 4 1 1 2 2 3 3 3 1 4 2 1 → G 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. 4 1 1 2 2 3 3 3 1 4 2 1 → G 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 χ e ( G ): minimum k such that G has a k -colouring. (Why χ e ?) ? 1 1 2 3 3 2 1 G χ e ( G ) ≤ 4. Equality?

Chromatic number χ e ( G ): minimum k such that G has a k -colouring. (Why χ e ?) ? 1 1 2 3 3 2 1 G χ e ( G ) ≤ 4. Equality? Yes.

Weak duality Theorem (Kostochka, Sopena, Zhu, 1997) Let G be an orientation of a graph with maximum degree ∆ . Then χ o ( G ) ≤ 2∆ 2 2 ∆ .

Weak duality Theorem (Kostochka, Sopena, Zhu, 1997) Let G be an orientation of a graph with maximum degree ∆ . Then χ o ( G ) ≤ 2∆ 2 2 ∆ . The statement also holds, with the same proof method, if − orientation is replaced by 2-edge-colouring, and − χ o is replaced by χ e .

Weak duality Theorem (Kostochka, Sopena, Zhu, 1997) Let G be an orientation of a graph with maximum degree ∆ . Then χ o ( G ) ≤ 2∆ 2 2 ∆ . The statement also holds, with the same proof method, if − orientation is replaced by 2-edge-colouring, and − χ o is replaced by χ e . This sort phenomenon happens frequently.

Weak duality Theorem (Kostochka, Sopena, Zhu, 1997) Let G be an orientation of a graph with maximum degree ∆ . Then χ o ( G ) ≤ 2∆ 2 2 ∆ . The statement also holds, with the same proof method, if − orientation is replaced by 2-edge-colouring, and − χ o is replaced by χ e . This sort phenomenon happens frequently. Not always: − Any orientation of P 5 has χ o ≤ 3; − An alternating 2-edge-colouring of P 5 has χ e = 4. 1 2 3 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 χ o ( G ) ≤ 2∆ 2 2 ∆ .

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 χ e ( G ) ≤ m ∆ 2 m ∆ .

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 ≤ S m , and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours of the edges incident with x according to γ .

Switching Let G be an m -edge-coloured graph, Γ ≤ S m , and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours of 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 ≤ S m , and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours of 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 ≤ S m , and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours of 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 ≤ S m , and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours of 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 ≤ S m , and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours of 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 ≤ S m , and γ ∈ Γ. For x ∈ V , switching at x with respect to γ permutes the colours of 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 of switches transforms G so that it becomes isomorphic to H .

Some groups might be less interesting Suppose Γ ≤ S m has Property T j : − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k . Using Property T j we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S 3 has this property for all j . Say j is red. y x

Some groups might be less interesting Suppose Γ ≤ S m has Property T j : − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k . Using Property T j we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S 3 has this property for all j . Say j is red. y x α = ( black red )( blue ) , β = ( red blue black )

Some groups might be less interesting Suppose Γ ≤ S m has Property T j : − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k . Using Property T j we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S 3 has this property for all j . Say j is red. y x α α = ( black red )( blue ) , β = ( red blue black )

Some groups might be less interesting Suppose Γ ≤ S m has Property T j : − it acts transitively, and − for all i there exists α ∈ Γ that sends i to j and fixes some k . Using Property T j we can make 4 switches that transform xy to colour j and leave all other colours unchanged. S 3 has this property for all j . Say j is red. y x α α = ( black red )( blue ) , β = ( red blue black )