Colouring, circular list colouring and adapted game colouring of - - PowerPoint PPT Presentation
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
Introduction 3-colourability of planar graphs Circular consecutive choosability of graphs 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
Under the supervision of
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
1
Introduction
2
3-colourability of planar graphs
3
Circular consecutive choosability of graphs
4
Adapted game colouring of graphs
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
5
: complete K
5
: cycle
C
4 , 3
: bipartite complete K graph planar r
tree
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 2 3 1
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)
1
A proper k-colouring of G is a mapping f : V(G) → {1, 2, · · · , k} such that f(x) = f(y) whenever xy is an edge of G.
2
A graph G is said to be k-colourable if G has a proper k-colouring.
3
The chromatic number of G, denote by χ(G), is defined as χ(G) = min{k : G is k-colourable}.
4
A graph G is called a k-chromatic graph if χ(G) = k.
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 2 2 2 2
2 ) ( = G χ
2 1 2 1 3
3 ) ( = G χ
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 Known results Configuration on HG
1
Introduction
2
3-colourability of planar graphs
3
Circular consecutive choosability of graphs
4
Adapted game colouring of graphs
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 Known results Configuration on HG
Theorem (Appel and Haken 1976) Every planar graph is 4-colourable. Theorem (Gr¨
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 Circular consecutive choosability of graphs Adapted game colouring of graphs Known results Configuration on HG
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 Circular consecutive choosability of graphs Adapted game colouring of graphs Known results Configuration on HG 1
Abbott and Zhou proved that such a k exists and k ≤ 11.
2
k ≤ 10 by Borodin.
3
k ≤ 9 by Borodin, Sanders and Zhao.
4
k ≤ 7 by Borodin, Glebov, Raspaud, and Salavatipour in 2005.
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 Known results Configuration on HG
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 Circular consecutive choosability of graphs Adapted game colouring of graphs Known results Configuration on HG
Definition (HG) For a planar graph G, let HG be the graph with vertex set V(HG) = {C : C is a cycle of G with |C| ∈ {4, 6, 7}} and E(HG) = {CiCj : Ci and Cj are adjacent in G}.
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 Known results Configuration on HG
:
1
G
1
x
2
x
3
x
4
x
5
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6
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7
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8
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9
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1
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2
C
3
C :
1G
H
). 1 ( ). 2 (
:
2
G
1
C
2
C
3
C :
2G
H
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8
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C :
3
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1
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). 3 (
8
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C
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 Known results Configuration on HG
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 HG 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 f0 is an i-face of G with 3 ≤ i ≤ 11. Then every proper 3-colouring of the vertices of f0 can be extended to the whole G.
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
1
Introduction
2
3-colourability of planar graphs
3
Circular consecutive choosability of graphs
4
Adapted game colouring of graphs
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
Definition (Circular Chromatic Number) Suppose r ≥ 1 is a real number, and G = (V, E) is a graph.
1
A circular r-colouring of G is a mapping f : V → S(r) such that for each edge xy of G, we have |f(x) − f(y)|r ≥ 1.
2
We say G is circular r-colourable if G has a circular r-colouring.
3
The circular chromatic number is defined as χc(G) = inf{r : G is circular r-colourable}.
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
A circular 2.5-colouring of C5
1
x
2
x
3
x
4
x
5
x 5 . 1 5 . 2 1
5
C
1
x
2
x
3
x
4
x
5
x 5 . 2 0 = 1 2 5 . 5 . 1 5 . 2 with ) ( = r r S 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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite 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).
→ v L :
) (r S
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
Definition (Circular Consecutive Choosability)
1
A graph G is called circular consecutive (t, r)-choosable if for any (t, r)-circular consecutive list assignment L, G has a circular L-colouring.
2
The circular consecutive choosability chr
cc(G) of G w.r.t r is
defined as chr
cc(G) = inf{t : G is circular consecutive (t, r)-choosable}.
3
The circular consecutive choosability of G is defined as chcc(G) = sup{chr
cc(G) : r ≥ χc(G)}.
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
Remark chcc(G) = inf{t : ∀r ≥ χc(G), G is cc (t, r)-choosable}
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
Lemma (Lin, Yang, Yang, Zhu) Suppose r′ > r. Then chr ′
cc(G) ≤ r ′ r chr cc(G) − r ′ r + 1.
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
Lemma (Lin, Yang, Yang, Zhu) If r = χc(G) and G has n vertices, then chr
cc(G) ≤ r(1 − 1/n)
Lemma (Lin, Yang, Yang, Zhu) For any r ≥ χc(G), ch2r
cc(G) ≤ chr cc(G)
Corollary For any graph G, chcc(G) = sup{chr
cc(G) : χc(G) ≤ r < 2χc(G)}
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
Theorem (Lin, Yang, Yang, Zhu) Suppose G is a graph on n vertices and r is a real number greater than or equal to χc(G). Then χ(G) − 1 ≤ chr
cc(G) ≤ r −
r |V(G)| − r χc(G) + 1 Corollary Suppose G is a graph on n vertices. Then χ(G) − 1 ≤ chcc(G) ≤ 2χc(G)(1 − 1/n) − 1.
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
Theorem (Lin, Yang, Yang, Zhu) Let T be a tree on n vertices. Then chcc(T) = 2(1 − 1
n)
Theorem (Lin, Yang, Yang, Zhu) For any integer n ≥ 1, chcc(Kn) = n − 1
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
Lemma (Lin, Yang, Yang, Zhu) For any integer n ≥ 3, chcc(Cn) ≥ 2 Lemma (Lin, Yang, Yang, Zhu)
1
If n ≥ 2 is even and r ≥ 2, then chr
cc(Cn) ≤ 2
2
If n is odd and r ≥ 3, then chr
cc(Cn) ≤ 2
Corollary If n ≥ 4 is even, then chcc(Cn) = 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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
Theorem (Liu) If n is odd and χc(Cn) ≤ r < 3, then chr
cc(Cn) ≤ 2
Corollary For any Cn, chcc(Cn) = 2 Theorem (Pan and Zhu) Every 2-choosable graph is circular consecutive 2-choosable
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 Circular chromatic number and circular choosability Some general bounds on chcc(G) Trees, cycles, complete graphs and complete bipartite graphs
Theorem (Lin, Yang, Yang, Zhu) For any real number r ∈ [2, 4) and any positive integer m, chr
cc(Km,m) = r
2 − r 2m + 1 Corollary For any m ≥ 1, chcc(Km,m) = 3 − 2 m
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
1
Introduction
2
3-colourability of planar graphs
3
Circular consecutive choosability of graphs
4
Adapted game colouring of graphs
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Definition (Adapted Colouring) Let G = (V, E) be a graph. Let F : E → N be an edge
G is adapted to F, if there is no edge e = uv such that f(u) = f(v) = F(e). (Here, all the colourings is not necessarily proper.)
1
x
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x
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x
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x
5
x
1
x
2
x
3
x
4
x
5
x 1 1 2 2 2 1 1 2 2 2
F f to adapted not is F f to adapted is '
2 2 2 1 1 2 2 1 1 1
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Definition (Game Chromatic Number)
1
Given a graph G and a colour set X.
2
Two players, Alice and Bob take turns colouring the vertices of G.
3
Bob has the first turn and Bob can pass his move.
4
Each partial colouring is a proper colouring.
5
Alice wins the game if all vertices are coloured. Or Bob wins the game.
6
The game chromatic number is denoted by χg(G) = min{|X|: Alice has a winning strategy}
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
G 1 2 1 3 ) ( = G
g
χ 2 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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
H 3 ) ( > H
g
χ 1 1 1 1 2 2 1 1 2 2 3 3
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Remark
1
Alice always wins if |X| = |V|
2
A subgraph may have much larger (or much smaller) game chromatic number
3
χ(G) ≤ χg(G) ≤ ∆(G) + 1 for all graph G
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Definition
1
Given a family G of graphs, let χg(G) = max{χg(G) : G ∈ G}.
2
F : the class of forests
3
Q : the class of outerplanar graphs
4
Tk : the class of k-trees
5
PT k : the class of partial k-trees
6
P : the class of planar graphs
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Theorem
1
χg(F) = 4
2
6 ≤ χg(Q) ≤ 7
3
2k + 1 ≤ χg(PT k) ≤ 3k + 2
4
8 ≤ χg(P) ≤ 17
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Definition (Adapted Game Chromatic Number)
1
Given a graph G, a edge colouring F, and a colour set X.
2
Two players, Alice and Bob take turns colouring the vertices of G.
3
Bob has the first turn and Bob can pass his move.
4
Each partial colouring is adapted to F.
5
Alice wins the game if all vertices are coloured. Or Bob wins the game.
6
The adapted game chromatic number is denoted by χadg(G) = min{|X|: Alice has a winning strategy}
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
2 1 2 1
4
P 3 ) (
4 =
P
adg
χ 2 1 2 1 1 1 2 1 2 1 2 1 2 1 2 1 1 1 2 2 1 2 1 2 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 2 1 2 1 2 1
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Theorem (Kierstead, Yang, Yang, Zhu) χadg(F) = 3 Alice’s strategy: Suppose Bob has just coloured a vertex x. (1) : If the father u = f(x) of x is uncoloured, then Alice colours u with any legal colour distinct from F(eu) (2) : If f(x) is already coloured or if Bob passed the move, then Alice colours either the root or any uncoloured vertex u′ whose father is already coloured
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
[the model of Alice’s strategy for trees:]
T
v
x ) (x f u =
u
e
x
e
Bob
) ( ) ( ) (
u x
e , F e F u c ≠
Alice
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Definition (Partial k-tree) A graph G is a k-tree if there is an ordering of the vertices of G, say v1, v2, · · · , vn, such that {v1, · · · , vk} induces a clique, and for each i ≥ k + 1, the set {vj : j < i, vi ∼ vj} induces a k-clique.
tree − 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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Theorem (Kierstead, Yang, Yang, Zhu) For any k-tree G, χadg(G) ≤ 2k + 1 G is a k-tree. F is an edge colouring. X is the colour set with 2k + 1 colours.
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
[the model of Alice’s strategy for k-trees:]
x u Y Y uncoloured
coloured Bob Alice uncoloured
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
There exists a k-tree G with χadg(G) ≥ k + 2.
2
v
3
v
leaves many v
1
v
4
v
5
v
6
v
6
leaves many v
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Corollary For any partial k-tree G, χadg(G) ≤ 2k + 1 Note that outerplanar graphs are partial 2-trees. Corollary If G is an outerplanar graph then χadg(G) ≤ 5 Proposition There exists an outerplanar graph G and an edge colouring F
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
1 2 1 2 1 2
4 4 4 4 4 4 4
3 3 3 3 3 3 3 3 3 3 3 3 2 1
1 2 1 2 1 2 1 2 1 2 1
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Theorem (Kierstead, Yang, Yang, Zhu) If G is a planar graph, then χadg(G) ≤ 11
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs i
v u =
I
coloured uncoloured
I
) (u N H
−
x
H B ⊂ O
Alice
H A ⊄ A D
vertex coloured last The
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs i
v u =
I
coloured uncoloured
I
) (u NH
−
x
H B ⊂
z
O
H A ⊄
A
) (z φ
|I| |A| | u |NH ≥ + −
−
1 ) (
Alice
vertex coloured last The
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Let B := O ∩ V(H) and A := O\B.
1
|N−
H (u) ∪ B| = |N− H (u)| + |B| ≤ 7
2
|N−
H (u)| − |A| + 1 ≥ |I|
Forbidden colours. |O − {x}| + |I| + |D| + |{x}| ≤ (|A| + |B| − 1) + (|N−
H (u)| − |A| + 1) + 3
≤ |N−
H (u)| + |B| + 3 ≤ 10
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Proposition There exists a planar graph G and an edge colouring F of G for which Bob can win the adapted game with 5 colours
5 R
1
G
2
G
5 5 5 5 5 5 5 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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs 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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Definition (Cartesian Product) Given two graphs G = (V, E) and G′ = (V ′, E′), the Cartesian product GG′ of G and G′ has vertex set V × V ′ = {(x, x′) : x ∈ V, x′ ∈ V ′}, and (x, x′) ∼ (y, y′) if either x = y and x′y′ ∈ E′ or xy ∈ E and x′ = y′
: G : H
H G
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Definition (Acyclic Colouring) An acyclic t-colouring is a mapping f : V(G) → {1, 2, · · · , t} s.t. (1) : ∀i, f −1(i) is an independent set (2) : ∀i = i′, the subgraph induced by f −1(i) ∪ f −1(i′) is a forest G is acyclic t-colourable if G has an acyclic t-colouring. The acyclic chromatic number χa(G) of G is defined as χa(G) = min{t : G is acyclic t-colourable}
4
C
1 2 2 1
colouring acyclic not
4
C
1 2 2
3 colouring acyclic
3 ) (
4 =
C
a
χ
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Theorem (Yang, Zhu) Let G be a graph with χa(G) ≤ t, and Tk be a k-tree. Then χadg(TkG) ≤ t(t + 2k) Corollary Let G be a graph with χa(G) ≤ t, and PTk be a partial k-tree. Then χadg(PTkG) ≤ t(t + 2k) Theorem (Yang, Zhu) Let G be a graph with χa(G) ≤ t, and P be a planar graph. Then χadg(PG) ≤ t(t + 10)
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
G
k
T
p
V u v ) ( ) ( u N v N
j j
U U + −
∩
A B uncoloured
) (u N
j
U −
D
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 Basic definitions and related concepts Partial k-trees Planar graphs Cartesian product graphs
Chung-Ying Yang Colouring, circular list colouring and adapted game colouring of gr