Backbone colouring of planar graphs Arnoosh Golestanian Brock - - PowerPoint PPT Presentation

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Backbone colouring of planar graphs Arnoosh Golestanian Brock - - PowerPoint PPT Presentation

Definition Previous Results Results Method Backbone colouring of planar graphs Arnoosh Golestanian Brock University (Joint project with Babak Farzad) . DMD/OCW 2015 Arnoosh Golestanian Backbone colouring of planar graphs Definition


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Definition Previous Results Results Method

Backbone colouring of planar graphs

Arnoosh Golestanian

Brock University (Joint project with Babak Farzad) .

DMD/OCW 2015

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Backbone colouring

A q-backbone k-colouring of (G, H) is a mapping f : V (G) → {1, 2, ..., k} such that: |f (u) − f (v)| ≥ q if uv ∈ E(H) |f (u) − f (v)| ≥ 1 if uv ∈ E(G) \ E(H)

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Backbone colouring

A q-backbone k-colouring of (G, H) is a mapping f : V (G) → {1, 2, ..., k} such that: |f (u) − f (v)| ≥ q if uv ∈ E(H) |f (u) − f (v)| ≥ 1 if uv ∈ E(G) \ E(H) Backbone chromatic number of (G, H) ⇒ The minimum number k for which there is a backbone k-colouring of (G, H) ⇒ BBCq(G, H).

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Backbone colouring

A q-backbone k-colouring of (G, H) is a mapping f : V (G) → {1, 2, ..., k} such that: |f (u) − f (v)| ≥ q if uv ∈ E(H) |f (u) − f (v)| ≥ 1 if uv ∈ E(G) \ E(H) Backbone chromatic number of (G, H) ⇒ The minimum number k for which there is a backbone k-colouring of (G, H) ⇒ BBCq(G, H). If q = 2 and H is a spanning tree T ⇒ BBC(G, T)

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Examples for BBC(G, T)

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1 2 3 2 5 3 1 2 1 4 Figure: k = 4, k = 5

∃ spanning tree T of G, k = 4 ⇒ BBC(G, T) = 4

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Circular Backbone colouring

A circular q-backbone k-colouring of (G, T) is a mapping f : V (G) → {1, 2, ..., k} such that: q ≤ |f (u) − f (v)| ≤ k − q if uv ∈ E(H) 1 ≤ |f (u) − f (v)| if uv ∈ E(G) \ E(H)

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Circular Backbone colouring

A circular q-backbone k-colouring of (G, T) is a mapping f : V (G) → {1, 2, ..., k} such that: q ≤ |f (u) − f (v)| ≤ k − q if uv ∈ E(H) 1 ≤ |f (u) − f (v)| if uv ∈ E(G) \ E(H) Circular backbone colouring of (G, H) is a special case of backbone k-colouring of (G, H).

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Circular Backbone colouring

A circular q-backbone k-colouring of (G, T) is a mapping f : V (G) → {1, 2, ..., k} such that: q ≤ |f (u) − f (v)| ≤ k − q if uv ∈ E(H) 1 ≤ |f (u) − f (v)| if uv ∈ E(G) \ E(H) Circular backbone colouring of (G, H) is a special case of backbone k-colouring of (G, H).

1 2 3 4 5 6 7

Figure: Circular colouring

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Circular Backbone colouring

Circular backbone chromatic number of (G, T) ⇒ The minimum number k for which there is a circular backbone k-colouring of (G, H) ⇒ CBCq(G, H).

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Circular Backbone colouring

Circular backbone chromatic number of (G, T) ⇒ The minimum number k for which there is a circular backbone k-colouring of (G, H) ⇒ CBCq(G, H). If q = 2 and H is a spanning tree T ⇒ CBC(G, T) If G has a backbone k-colouring for any spanning tree T, then G has a circular backbone k + 1-colouring.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Examples for CBC(G, T)

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3 1 2 1 4

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1 2 3 2 5 Figure: k = 5, k′ = 6

∃ spanning tree T of G, k = 5 ⇒ CBC(G, T) = 5.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

History

Four colour theorem: (Apel-Haken; 1977) If G is planar, then χ(G) ≤ 4; every plane map is 4-colourable.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

History

Four colour theorem: (Apel-Haken; 1977) If G is planar, then χ(G) ≤ 4; every plane map is 4-colourable. Three-colour theorem: (Grotzsh; 1959) If G is planar and triangular free, then χ(G) ≤ 3.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

History

Four colour theorem: (Apel-Haken; 1977) If G is planar, then χ(G) ≤ 4; every plane map is 4-colourable. Three-colour theorem: (Grotzsh; 1959) If G is planar and triangular free, then χ(G) ≤ 3. Steinberg conjecture: (Borodin; 1976) Every {4, 5}-cycle free planar graph is 3-colourable.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Steinberg-like theorems for backbone colouring

(Broersma, Fomin, Golovach, Woeginger; 2003) For any connected graph G and spanning tree T, BBC(G, T) ≤ 2χ(G) − 1.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Steinberg-like theorems for backbone colouring

(Broersma, Fomin, Golovach, Woeginger; 2003) For any connected graph G and spanning tree T, BBC(G, T) ≤ 2χ(G) − 1. Four-Colour Theorem ⇒ if G is a planar graph, then BBC(G, T) ≤ 7 and BBCq(G, H) ≤ 3q + 1.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Steinberg-like theorems for backbone colouring

(Broersma, Fomin, Golovach, Woeginger; 2003) For any connected graph G and spanning tree T, BBC(G, T) ≤ 2χ(G) − 1. Four-Colour Theorem ⇒ if G is a planar graph, then BBC(G, T) ≤ 7 and BBCq(G, H) ≤ 3q + 1. (Campos, Havet, Sampaio and Silva; 2013) If T has diameter at most 4 then BBC(G, T) ≤ 6.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Steinberg-like theorems for backbone colouring

(Havet, King, Liedloff and Todinca; 2014) For a planar graph G, BBCq(G, H) ≤ q + 6 and BBC3(G, H) ≤ 8.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Steinberg-like theorems for backbone colouring

(Havet, King, Liedloff and Todinca; 2014) For a planar graph G, BBCq(G, H) ≤ q + 6 and BBC3(G, H) ≤ 8. (Bu and Zhang; 2011) If G is a connected C4-free planar graph, BBC(G, T) ≤ 4.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Steinberg-like theorems for backbone colouring

(Havet, King, Liedloff and Todinca; 2014) For a planar graph G, BBCq(G, H) ≤ q + 6 and BBC3(G, H) ≤ 8. (Bu and Zhang; 2011) If G is a connected C4-free planar graph, BBC(G, T) ≤ 4. (Zhang and Bu; 2010) If G is a connected non-bipartite C5-free planar graph, BBC(G, T) = 4.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Steinberg-like theorems for backbone colouring

(Bu and Li; 2011) If G is a connected C6-free or C7-free planar graphs without adjacent triangles, BBC(G, T) ≤ 4.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Steinberg-like theorems for backbone colouring

(Bu and Li; 2011) If G is a connected C6-free or C7-free planar graphs without adjacent triangles, BBC(G, T) ≤ 4. (Wang; 2012) If G is a connected planar graph without C8 and adjacent triangles, BBC(G, T) ≤ 4.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Steinberg-like theorems for backbone colouring

(Havet, King, Liedloff and Todinca; 2014) CBCq(G, H) ≤ qχ(G) and CBCq(G, H) ≤ 2q + 4.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Steinberg-like theorems for backbone colouring

(Havet, King, Liedloff and Todinca; 2014) CBCq(G, H) ≤ qχ(G) and CBCq(G, H) ≤ 2q + 4. (Araujo, Havet, Schmitt; 2014) If G is a planar graph containing no cycle on 4 or 5 vertices and H ⊆ G is a forest, then CBC(G, H) ≤ 7.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method

Backbone colourings of planar graphs without adjacent triangles

If G is a connected planar graph without adjacent triangles, then there exists a spanning tree T of G such that BBC(G, T) ≤ 4. If G is a connected C4-free and C5-free planar graph, then for every spanning tree T of G, CBC(G, T) ≤ 7.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Minimum counterexample

Proof by contradiction and G(V , E) is a minimum counterexample with minimum |V |. If graph G ′(V ′, E ′) is a connected planar graph without adjacent triangles and |V ′| ≤ |V | ⇒ BBC(G ′, T ′) ≤ 4.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Minimum counterexample

Proof by contradiction and G(V , E) is a minimum counterexample with minimum |V |. If graph G ′(V ′, E ′) is a connected planar graph without adjacent triangles and |V ′| ≤ |V | ⇒ BBC(G ′, T ′) ≤ 4. If f and f ′ are two vertex colouring of G and ∀v ∈ V (G), f ′(v) + f (v) = k + 1, then f is a backbone k-colouring of (G, T) ↔ f ′ is a backbone k-colouring of (G, T).

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Properties of minimum counterexample

Minimum degree of counterexample: δ(G) ≥ 4.

  • Arnoosh Golestanian

Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Properties of minimum counterexample

Minimum degree of counterexample: δ(G) ≥ 4. If x, y and z are three mutually adjacent 4-vertices in G, then x, y, z form a bad-triangle.

  • Arnoosh Golestanian

Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Properties of minimum counterexample

Minimum degree of counterexample: δ(G) ≥ 4. If x, y and z are three mutually adjacent 4-vertices in G, then x, y, z form a bad-triangle.

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y z

1 1 1 2 2

x y z x y

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z

Figure: Three mutually adjacent 4-vertices.

list L(v) is a set of available colours for vertex v.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Properties of counterexample

Set A = {x, y, z} If ∃u, v ∈ A, L(u) − L(v) = ∅.

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2 4 1 2 2 3 3 4 x y z

Figure: An example for finding spanning tree T.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Properties of counterexample

L(x) = L(y) = L(z) = {c1, c2}. Construct a graph G ′′ by adding two new edges to G/x, y, z which does not create any adjacent triangles in G ′′.

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y z

1 1 1 1 2 2 2 2 3

x y z w w w y z x

Figure: An example for graph G ′′.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

bad-triangle

When by adding any two edges to G/x, y, z, where BBC(G ′′, T ′′) ≥ 5, adjacent triangles are created, bad-triangle.

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y z

1 1 1 1 2 2 2 2 3

x y z w w w y z x

Figure: bad-triangles.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

bad-triangle

Set B = N(A) − A, i.e. B = {x1, x2, y1, y2, z1, z2}. vertices in set B are sponsor vertices of bad-triangle. Degree of sponsor vertices is at least 5.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Discharging Rules

Initial charge function is ω(x) = d(x) − 4, x ∈ V (G) ∪ F(G).

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Discharging Rules

Initial charge function is ω(x) = d(x) − 4, x ∈ V (G) ∪ F(G). The distribution of charges is according to the defined discharging rules.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Discharging Rules

Initial charge function is ω(x) = d(x) − 4, x ∈ V (G) ∪ F(G). The distribution of charges is according to the defined discharging rules. When discharging is in progress, the sum of all charges is fixed.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Nonnegative Charge

ω′ is the new charge of vertices and faces.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Nonnegative Charge

ω′ is the new charge of vertices and faces. ∀x ∈ V (G) ∪ F(G), ω′(x) ≥ 0.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Nonnegative Charge

ω′ is the new charge of vertices and faces. ∀x ∈ V (G) ∪ F(G), ω′(x) ≥ 0. By using Euler’s formula, (d(v) − 4) + (d(f ) − 4) = −8, contradiction.

Arnoosh Golestanian Backbone colouring of planar graphs

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Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Thank you.

Arnoosh Golestanian Backbone colouring of planar graphs