Embed Size (px)
Citation preview
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
2/06/2015 12:57
Considered problems: list coloring
2/06/2015 12:57
Choosability
2/06/2015 12:57
Choosability: first remarks
2/06/2015 12:57
First examples in grid graphs
2/06/2015 12:57
First examples in grid graphs
2/06/2015 12:57
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
2/06/2015 13:27
1,3 2,3
1,2 1,2
1,3 2,3
First examples in grid graphs
2/06/2015 13:00
1,3 2,3
1,2 1,2
1,3 2,3
First examples in grid graphs
2/06/2015 12:58
1,3 2,3
1,2 1,2
1,3 2,3
First examples in grid graphs
2/06/2015 12:58
Grids are not (2,3)-choosable
2/06/2015 12:57
(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
2/06/2015 12:57
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
2/06/2015 12:58
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 12:58
Characterization of 2-choosability
2/06/2015 12:58
Characterization of 2-choosability
2/06/2015 12:58
Path with an even
number of ver?ces
Path with an odd
number of ver?ces
Characterization of 2-choosability
2/06/2015 12:58
Sketch of proof:
1
Characterization of 2-choosability
2/06/2015 12:58
Sketch of proof (cont.):
2
Characterization of 2-choosability
2/06/2015 12:58
Sketch of proof (cont.):
3
Characterization of 2-choosability
2/06/2015 12:58
Sketch of proof (cont.):
3
2/06/2015 12:58
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
2/06/2015 12:58
Complexity of choosability Which relevant complexity class?
2/06/2015 12:58
Complexity of choosability Main known results
2/06/2015 12:58
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 12:58
Choosability with fixed number of colors
2/06/2015 12:58
(MD., D. de Werra, 2015)
Grids are not [(2,3),4]-choosable
2/06/2015 12:58
(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
2/06/2015 12:58
Characterization of 2-choosability (remind …)
2/06/2015 12:58
2-choosability revisited
2/06/2015 12:58
2-choosability revisited
2/06/2015 13:33
k-choosability vs (k+1)-choosability
2/06/2015 12:58
k-choosability vs (k+1)-choosability
2/06/2015 13:33
Complexity with few colors
2/06/2015 12:58
Complexity with few colors
2/06/2015 13:34
Complexity with few colors
2/06/2015 13:34
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?
12:58
