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 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

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

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

Definition Previous Results Results Method

Examples for BBC(G, T)

1

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

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

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

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

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

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

Definition Previous Results Results Method

Examples for CBC(G, T)

1

3 1 2 1 4

1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

• x

y z

1 1 1 2 2

x y z x y

2

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

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) = ∅.

• 1

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

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 ′′.

• x

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

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.

• x

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

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

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

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

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

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

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

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

Definition Previous Results Results Method Properties of minimum counterexample Discharging Method

Thank you.

Arnoosh Golestanian Backbone colouring of planar graphs