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Introduction 3-colourability of planar graphs Circular consecutive choosability of graphs Adapted game colouring of graphs Colouring, circular list colouring and adapted game colouring of graphs Chung-Ying Yang August 8th, 2010 Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Circular consecutive choosability of graphs Adapted game colouring of graphs Acknowledgment Under the supervision of Professor Xuding Zhu Joint works with Professor: Wensong Lin, Daqing Yang and H. A. Kierstead Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Circular consecutive choosability of graphs Adapted game colouring of graphs Introduction 1 3-colourability of planar graphs 2 Circular consecutive choosability of graphs 3 Adapted game colouring of graphs 4 Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Circular consecutive choosability of graphs Adapted game colouring of graphs K K C complete : 5 - cycle : complete bipartite : 5 3 , 4 5 tree planar graph outerplana r Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Circular consecutive choosability of graphs Adapted game colouring of graphs Definition A proper colouring of a graph is an assignment of ”colours” to the elements of the graph such that adjacent elements get different colours. 1 3 1 2 Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Circular consecutive choosability of graphs Adapted game colouring of graphs Definition ( Chromatic number) A proper k-colouring of G is a mapping 1 f : V ( G ) → { 1 , 2 , · · · , k } such that f ( x ) � = f ( y ) whenever xy is an edge of G. A graph G is said to be k-colourable if G has a proper 2 k-colouring. The chromatic number of G, denote by χ ( G ) , is defined as 3 χ ( G ) = min { k : G is k -colourable } . A graph G is called a k-chromatic graph if χ ( G ) = k. 4 Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Circular consecutive choosability of graphs Adapted game colouring of graphs 1 1 1 2 2 3 1 2 2 2 2 = G = χ G χ ( ) 3 ( ) 2 Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Known results Circular consecutive choosability of graphs Configuration on H G Adapted game colouring of graphs Introduction 1 3-colourability of planar graphs 2 Circular consecutive choosability of graphs 3 Adapted game colouring of graphs 4 Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Known results Circular consecutive choosability of graphs Configuration on H G Adapted game colouring of graphs Four-Colour Theorem Theorem ( Appel and Haken 1976) Every planar graph is 4 -colourable. Theorem ( Gr¨ otzsch’s Theorem 1959) Every triangle-free planar graph is 3 -colourable. Theorem ( Garey, Johnson and Stockmeyer 1976) Deciding a planar graph is 3-colourable is NP-complete. Conjecture ( Steinberg’s Conjecture 1976) Every planar graph without 4 - and 5 -cycles is 3 -colourable. Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Known results Circular consecutive choosability of graphs Configuration on H G Adapted game colouring of graphs Conjecture ( Relaxation of Steinberg’s Conjecture) Find the minimum k, if it exists, s.t. every planar graph without cycles of length ℓ for 4 ≤ ℓ ≤ k is 3 -colourable. Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Known results Circular consecutive choosability of graphs Configuration on H G Adapted game colouring of graphs Abbott and Zhou proved that such a k exists and k ≤ 11. 1 k ≤ 10 by Borodin. 2 k ≤ 9 by Borodin, Sanders and Zhao. 3 k ≤ 7 by Borodin, Glebov, Raspaud, and Salavatipour in 4 2005. Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Known results Circular consecutive choosability of graphs Configuration on H G Adapted game colouring of graphs Theorem ( Borodin, Glebov, Raspaud, and Salavatipour) Every planar graph without cycles of length from 4 ∼ 7 is 3-colourable. Theorem ( Borodin, Montassier and Raspaud) Planar graphs without adjacent cycles of length 3 ∼ 7 are 3 -colourable. Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Known results Circular consecutive choosability of graphs Configuration on H G Adapted game colouring of graphs Definition ( H G ) For a planar graph G, let H G be the graph with vertex set V ( H G ) = { C : C is a cycle of G with | C | ∈ { 4 , 6 , 7 }} and E ( H G ) = { C i C j : C i and C j are adjacent in G } . Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Known results Circular consecutive choosability of graphs Configuration on H G Adapted game colouring of graphs x 8 x x x C 1 2 3 3 x 9 G H ( 1 ). : : G 1 1 x x x C C 6 5 4 1 2 x 7 x C C x 9 4 5 8 x x 10 7 x 11 x x G 2 3 ( 2 ). x H C : : 2 1 G 3 2 x 12 C C 1 2 x x x 6 5 4 x x 9 10 x C x x x 3 3 1 2 8 C C 1 4 G H ( 3 ). : : 3 G x x x x 3 7 6 5 4 C 2 x x 12 11 Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction 3-colourability of planar graphs Known results Circular consecutive choosability of graphs Configuration on H G Adapted game colouring of graphs Theorem ( Yang and Zhu) For a planar graph G, if any 3 -cycles and 5 -cycles are not adjacent to i-cycles whenever 3 ≤ i ≤ 7 , and H G is a forest, then G is 3 -colourable. Let Ω be the set of connected planar graphs satisfying the assumption of this theorem. Lemma ( Extend-Lemma) Suppose G ∈ Ω and f 0 is an i-face of G with 3 ≤ i ≤ 11 . Then every proper 3 -colouring of the vertices of f 0 can be extended to the whole G. Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction Circular chromatic number and circular choosability 3-colourability of planar graphs Some general bounds on ch cc ( G ) Circular consecutive choosability of graphs Trees, cycles, complete graphs and complete bipartite graphs Adapted game colouring of graphs Introduction 1 3-colourability of planar graphs 2 Circular consecutive choosability of graphs 3 Adapted game colouring of graphs 4 Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction Circular chromatic number and circular choosability 3-colourability of planar graphs Some general bounds on ch cc ( G ) Circular consecutive choosability of graphs Trees, cycles, complete graphs and complete bipartite graphs Adapted game colouring of graphs Definition ( Circular Chromatic Number) Suppose r ≥ 1 is a real number, and G = ( V , E ) is a graph. A circular r-colouring of G is a mapping f : V → S ( r ) such 1 that for each edge xy of G, we have | f ( x ) − f ( y ) | r ≥ 1 . We say G is circular r-colourable if G has a circular 2 r-colouring. The circular chromatic number is defined as 3 χ c ( G ) = inf { r : G is circular r -colourable } . Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction Circular chromatic number and circular choosability 3-colourability of planar graphs Some general bounds on ch cc ( G ) Circular consecutive choosability of graphs Trees, cycles, complete graphs and complete bipartite graphs Adapted game colouring of graphs A circular 2 . 5-colouring of C 5 0 = 2 . 5 0 x 1 x 2 x 1 x x x 0 . 5 1 4 5 2 3 1 . 5 x x 4 3 x x 2 5 0 . 5 2 1 . 5 1 C = S r r ( ) with 2 . 5 5 Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr

Introduction Circular chromatic number and circular choosability 3-colourability of planar graphs Some general bounds on ch cc ( G ) Circular consecutive choosability of graphs Trees, cycles, complete graphs and complete bipartite graphs Adapted game colouring of graphs Definition ( ( t , r ) -circular consecutive list assignment) A ( t , r ) -circular consecutive list assignment is function L that assigns each vertex a closed interval of length t of S ( r ) . → L : v S ( r ) Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr