On choosability of graphs with limited number of...
Transcript of On choosability of graphs with limited number of...
On choosability of graphs with limited number of colours
M. Demange, RMIT University, Melbourne, Vic -‐ Australia [email protected]
In collabora?on with D. De Werra, Ecole Polytechnique Fédérale de Lausanne, Switzerland [email protected]
2/06/2015 12:57
Discrete Maths Research Group mee?ng – Monash University – May 25, 2015
Outline
• A star?ng example
• Choosability: defini?ons and first examples • Some classes of choosable graphs
• Complexity of choosability
• Choosability with few colors
2/06/2015 12:57
Outline
• A star-ng example
• Choosability: defini?ons and first examples • Some classes of choosable graphs
• Complexity of choosability
• Choosability with few colors
2/06/2015 12:57
1 to 6 5 to 8 3 to 7 2 to 4 7 to 9
?
3 rooms 5 requirements
Booking system: coloring interval graphs
Pink Green Blue
1 to 6 5 to 8 3 to 7 2 to 4 6 to 9
5 to 8 3 to 7 2 to 4 7 to 9
Booking system
Pink Green Blue
1 to 6 5 to 8 3 to 7 2 to 4 6 to 9
1 to 6 5 to 8 3 to 7 2 to 4 6 to 9
Pink Green Blue
Booking system
1 to 6 5 to 8 3 to 7 2 to 4 7 to 9
Pink Green Blue
Booking system
1 to 6 5 to 8 3 to 7 2 to 4 7 to 9
1 2 3 4 5 6 7 8 9
Interval graph
Chromatic number = 3
M. Demange - RMIT
Polynomial
1 to 6 5 to 8 3 to 7 2 to 4 7 to 9
3 rooms 5 requirements
Each traveler can propose choices: list-coloring
Pink Green Blue
1 to 6 5 to 8 3 to 7 2 to 4 7 to 9
M. Demange - RMIT
1 to 6 5 to 8 3 to 7 2 to 4 7 to 9
M. Demange - RMIT
1 to 6 5 to 8 3 to 7 2 to 4 7 to 9
M. Demange - RMIT
Outline
• A star?ng example
• Choosability: defini-ons and first examples • Some classes of choosable graphs
• Complexity of choosability
• Choosability with few colors
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Considered problems: list coloring
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Choosability
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Choosability: first remarks
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First examples in grid graphs
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First examples in grid graphs
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1,3 2,3
1,2 1,2
1,3 2,3
First examples in grid graphs
2/06/2015 13:27
1,3 2,3
1,2 1,2
1,3 2,3
First examples in grid graphs
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1,3 2,3
1,2 1,2
1,3 2,3
First examples in grid graphs
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1,3 2,3
1,2 1,2
1,3 2,3
First examples in grid graphs
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1,3 2,3
1,2 1,2
1,3 2,3
First examples in grid graphs
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Grids are not (2,3)-choosable
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(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
Grids are not (2,3)-choosable
2/06/2015 13:28
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
Grids are not (2,3)-choosable
2/06/2015 13:19
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
Grids are not (2,3)-choosable
2/06/2015 13:03
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
Grids are not (2,3)-choosable
2/06/2015 13:11
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
Grids are not (2,3)-choosable
2/06/2015 13:10
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
2,1 1,2,4
2,3,4 1,3
1,2,3
1,4
1,2,4
1,2
2,3,4
1,3
1,2,3
1,4
Grids are not (2,3)-choosable
2/06/2015 13:13
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
2,1 1,2,4
2,3,4 1,3
1,2,3
1,4
1,2,4
1,2
2,3,4
1,3
1,2,3
1,4
Grids are not (2,3)-choosable
2/06/2015 13:20
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
2,1 1,2,4
2,3,4 1,3
1,2,3
1,4
1,2,4
1,2
2,3,4
1,3
1,2,3
1,4
Grids are not (2,3)-choosable
2/06/2015 13:21
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
2,1 1,2,4
2,3,4 1,3
1,2,3
1,4
1,2,4
1,2
2,3,4
1,3
1,2,3
1,4
Grids are not (2,3)-choosable
2/06/2015 13:01
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
2,1 1,2,4
2,3,4 1,3
1,2,3
1,4
1,2,4
1,2
2,3,4
1,3
1,2,3
1,4
Grids are not (2,3)-choosable
2/06/2015 13:22
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
2,1 1,2,4
2,3,4 1,3
1,2,3
1,4
1,2,4
1,2
2,3,4
1,3
1,2,3
1,4
Grids are not (2,3)-choosable
2/06/2015 13:01
(MD, de Werra, 2013)
X
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
2,1 1,2,4
2,3,4 1,3
1,2,3
1,4
1,2,4
1,2
2,3,4
1,3
1,2,3
1,4
Grids are not (2,3)-choosable
2/06/2015 13:25
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
2,1 1,2,4
2,3,4 1,3
1,2,3
1,4
1,2,4
1,2
2,3,4
1,3
1,2,3
1,4
Grids are not (2,3)-choosable
2/06/2015 13:26
(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
2,1 1,2,4
2,3,4 1,3
1,2,3
1,4
1,2,4
1,2
2,3,4
1,3
1,2,3
1,4
Grids are not (2,3)-choosable
2/06/2015 13:30
(MD, de Werra, 2013)
? Find an example with less than 41 ver?ces
1
4
2
5
3
6
10
7
11
8
12
9
13
16
14
17
15
18
22
19
23
20
24
21
25
28
26
29
27
30
34
31
35
32
36
33
37
40
38
41
39
42
Grids are (2,4)-choosable
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Acyclic orienta?on s.t.
2
1
,3)(
,1)(
VxxdVxxd
∈≤
∈≤+
+
Number ver?ces by elimina?ng Ver?ces s.t. 0)( =+ xd
(MD, de Werra, 2013)
2/06/2015 12:58
Grids are 3-choosable
Choosability of bipartite graphs
Theorem (Alon, Tarsi, Combinatorica 12, 1992) A bipartite graph of maximum degree is -choosable Δ ⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎥⎦
⎥⎢⎣
⎢Δ+⎥⎥
⎤⎢⎢
⎡Δ 12
,12
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Outline
• A star?ng example
• Choosability: defini?ons and first examples • Some classes of choosable graphs
• Complexity of choosability
• Choosability with few colors
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Characterization of 2-choosability
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Characterization of 2-choosability
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Path with an even
number of ver?ces
Path with an odd
number of ver?ces
Characterization of 2-choosability
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Sketch of proof:
1
Characterization of 2-choosability
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Sketch of proof (cont.):
2
Characterization of 2-choosability
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Sketch of proof (cont.):
3
Characterization of 2-choosability
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Sketch of proof (cont.):
3
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Choosability of planar graphs
Planar + no 3- and 4-cycles:3-choosable
Planar+triangle free:4-choosable
Planar: 4-colorable (four colors theorem, Appel and Haken 1977)
Triangle-free planar: 3-colorable (Grotzsch’s theorem, 1959)
Planar: 5-choosable
Planar+bipartite:3-choosable
Outline
• A star?ng example
• Choosability: defini?ons and first examples • Some classes of choosable graphs
• Complexity of choosability
• Choosability with few colors
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Complexity of choosability Which relevant complexity class?
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Complexity of choosability Main known results
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Outline
• A star?ng example
• Choosability: defini?ons and first examples • Some classes of choosable graphs
• Complexity of choosability
• Choosability with few colors
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Choosability with fixed number of colors
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(MD., D. de Werra, 2015)
Grids are not [(2,3),4]-choosable
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(MD, de Werra, 2013)
1,2 1,2,3
2,3 1,2,3
1,2,4 4,3
4,2
1,3,4
1,3
2,1 1,2,4
2,3,4 1,3
1,2,3
1,4
1,2,4
1,2
2,3,4
1,3
1,2,3
1,4
Bipartite graphs are [(2,3),3]- choosable
2-choosability revisited
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Characterization of 2-choosability (remind …)
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2-choosability revisited
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2-choosability revisited
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k-choosability vs (k+1)-choosability
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k-choosability vs (k+1)-choosability
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Complexity with few colors
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Complexity with few colors
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Complexity with few colors
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Complexity with few colors: [3,k]-choosability
No list coloring
Complexity with few colors: [3,k]-choosability
No list coloring
Complexity with few colors: [3,k]-choosability , 3
3,
, 3
3,
3 will never be used
No list coloring
Complexity with few colors: [3,k]-choosability , 3
3,
, 3
3,
3 will never be used
No list coloring
Complexity with few colors: triangle-free planar graphs
No list coloring
Complexity with few colors: triangle-free planar graphs
Color 3 forbidden for A
Last remark
An exercise: is this graph 2-choosable?
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