# An improved fixed-parameter algorithm for vertex cover

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```					Acyclic colorings of subcubic
graphs

San Skulrattanakulchai
Information Processing Letters 92 (2004) 161-167

報告人:葉曉宏
Definition
   subcubic graph
– vV, deg(v)≦3
   acyclic coloring
   acyclic edge coloring
3           1
1           1
2        1   2        3

2        2   2       2
1        2   1        2
1           3
X                           X           
Acyclic Coloring
At most 1 neighbor of x is colored
u

x
1   v        w

x has exactly 2 colored neighbors
u                        u

x             2
x       2
v            w                v       w
1           2                1       1
3                  3
Acyclic Coloring (cont.)
x has 3 colored neighbors
2           4
case 1                      case 2

u   1                              u   1

x                   2
x               4
v            w                  v               w
2            3                  1 4                 1
3                              3
Acyclic Coloring (cont.)
x has 3 colored neighbors (cont.)
case 3
u   2

3
x            3
v             w
4        1 2           12   4
Acyclic Edge Coloring
At most 1 neighbor of e is colored

1

e has exactly 2 colored neighbors

2   3       2   3
1                                   1   1

2
Acyclic Edge Coloring (cont.)
e has exactly 3 colored neighbors
4   3       4   3
1            3                     1   1

2                                  2

e has exactly 4 colored neighbors
case 1

1            3

2            4
Acyclic Edge Coloring (cont.)
e has exactly 4 colored neighbors (cont.)
case 2

3     4              3    4        3    4        3   4
15        1                       15   1

2         2                       2    2
5     1              5     1       5    4        5   4
Acyclic Edge Coloring (cont.)
e has exactly 4 colored neighbors (cont.)
case 3

4     5              4     5
1 3     2 1

2         3
Reference
[1] B. Grünbaum, Acyclic colorings of planar graphs, Israel J. Math. 14 (1973) 390–408.
[2] A.V. Kostochka, Upper bounds of chromatic functions of graphs, Ph.D. Thesis, Novosibirsk, 1978 (in
Russian).
[3] N. Alon, C.J.H. McDiarmid, B.A. Reed, Acyclic coloring of graphs, Random Structures and Algorithms 2
(1991) 277–288.
[4] G. Fertin, E. Godard, A. Raspaud, Minimum feedback vertex set and acyclic coloring, Inform. Process.
Lett. 84 (2002) 131–139.
[5] O.V. Borodin, On acyclic colourings of planar graphs, Discrete Math. 25 (1979) 211–236.
[6] O.V. Borodin, A.V. Kostochka, D.R. Woodall, Acyclic colouring of planar graphs with large girth, J.
London Math. Soc. 60 (1999) 344–352.
[7] O.V. Borodin, A.V. Kostochka, A. Raspaud, E. Sopena, Acyclic colouring of 1-planar graphs, Diskretnyi
Analiz i Issledovanie Operacii, Series 1 6 (4) (1999) 20–35.
[8] O.V. Borodin, D.G. Fon-Der Flaass, A.V. Kostochka, A. Raspaud, E. Sopena, Acyclic list 7-coloring of
planar graphs, J. Graph Theory 40 (2) (2002) 83–90.
[9] N. Alon, B. Sudakov, A. Zaks, Acyclic edge colorings of graphs, J. Graph Theory 37 (2001) 157–167.
[10] M.I. Burnstein, Every 4-valent graph has an acyclic 5-coloring, Soobš Akad. Nauk Gruzin SSR 93 (1979)
21–24 (in Russian).
[11] N. Alon, A. Zaks, Algorithmic aspects of acyclic edge colorings, Algorithmica 32 (2002) 611–614.

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