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 ) ⇒ BBC q ( 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 ) ⇒ BBC q ( 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 ) 2 5 1 2 4 1 3 3 2 1 1 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 7 2 6 3 5 4 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 ) ⇒ CBC q ( 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 ) ⇒ CBC q ( 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 ) 2 5 1 2 4 1 3 3 2 1 1 1 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 ) ≤ 2 χ ( 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 ) ≤ 2 χ ( G ) − 1. Four-Colour Theorem ⇒ if G is a planar graph, then BBC ( G , T ) ≤ 7 and BBC q ( G , H ) ≤ 3 q + 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 ) ≤ 2 χ ( G ) − 1. Four-Colour Theorem ⇒ if G is a planar graph, then BBC ( G , T ) ≤ 7 and BBC q ( G , H ) ≤ 3 q + 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 , BBC q ( G , H ) ≤ q + 6 and BBC 3 ( 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 , BBC q ( G , H ) ≤ q + 6 and BBC 3 ( G , H ) ≤ 8. (Bu and Zhang; 2011) If G is a connected C 4 -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 , BBC q ( G , H ) ≤ q + 6 and BBC 3 ( G , H ) ≤ 8. (Bu and Zhang; 2011) If G is a connected C 4 -free planar graph, BBC ( G , T ) ≤ 4. (Zhang and Bu; 2010) If G is a connected non-bipartite C 5 -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 C 6 -free or C 7 -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 C 6 -free or C 7 -free planar graphs without adjacent triangles, BBC ( G , T ) ≤ 4. (Wang; 2012) If G is a connected planar graph without C 8 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) CBC q ( G , H ) ≤ q χ ( G ) and CBC q ( G , H ) ≤ 2 q + 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) CBC q ( G , H ) ≤ q χ ( G ) and CBC q ( G , H ) ≤ 2 q + 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 C 4 -free and C 5 -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 Properties of minimum counterexample Results Discharging Method 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