the equitable coloring of kneser graphs 陳伯亮＆黃國卿 2008 年 8 月 11 日

The Equitable Coloring of Kneser Graphs

陳伯亮＆黃國卿

2008 年 8 月 11 日

• A proper k-coloring of a graph G is an labeling

f： V(G) {1,2,...,k} such that adjacent verti

ces have different labels. The labels are colors;

The vertices of one color form a color class.

• A graph G is k-colorable if G has a proper k-co

loring.

• The chromatic number of a graph G, denoted b

y , is the least k such that G is k-colorab

le.

)(G

• A equitable k-coloring of a graph G is an prop

er k-coloring f： V(G) {1,2,...,k} such that

||f -1(i)|-|f -1(j)|| 1 for all 1 i j k.

• A graph G is equitably k-colorable if G has a e

quitable k-coloring.

• The equitable chromatic number of a graph G,

denoted by , is the least k such that G is

equitably k-colorable.

• The equitable chromatic threshold of a graph

G, denoted by , is the least k such tha

t G is equitably n-colorable for all n k.

)(G

)(* G

• Lemma.

)()()( * GGG

• If graph G is equitably k-colorable, then the size of al

l color classes in a nonincreasing sort will be

• or the sizes of all color classes in a nondecreasing sor

t will be

k

kGV

k

GV

k

GV 1)(...,,

1)(,

)(

,1)(

...,,1)(

,)(

k

kGV

k

GV

k

GV

K3,3

2)( 3,3 K 2)( 3,3 K 4)( 3,3* K

K5,8

2)( 8,5 K

K5,8

3)( 8,5 K2)( 8,5 K

K5,8

5)( 8,5* K3)( 8,5 K2)( 8,5 K

• Theorem.

.

• Theorem. (Hajnal and Szemerédi； 1970)

.1)()(* GG

1)()( GG

• Lemma：

1)()()()( * GGGG

• Theorem. (Brooks； 1964)

Let G be a connected graph. Then

if )()( GG 12 nn CGandKG

• Conjecture. (Meyer； 1973)

Let G be a connected graph. Then

if )()( GG 12 nn CGandKG

• Conjecture. (Chen, Lih and Wu； 1994)

A connected graph G is equitable (G)-c

olorable if and only if

12,1212, nnnn KGandCGKG

• Theorem. (Guy； 1975)

A tree T is equitably k-colorable if k

• Theorem. (Bollobas and Guy ； 1983)

A tree T is equitably 3-colorable if

12

)(

T

10)(3)(8)(3)( TTVorTTV

• Theorem. (Chen and Lih ； 1994)

A tree T = T(X,Y), if and only

if

If , then

2)( T

.1|||| YX

2)( T .2)(* T

• Theorem. (Chen and Lih ； 1994)

Let T be a tree such that , then

, where v is an arbitrary major vertex.

2)( T

2])[\(

1||,3max)()( *

vNT

TTT

• Theorem. (Wu ； 1994)

is equitably k-colorable if and on

ly if

and for all i, where

tnnnK ,...,, 21

t

i

it

i

i

kn

nk

kn

n

11 //

k

n

kn

nni /

.1

t

iinn

• For n 2k+1, the Kneser graph KG(n,k) has

the vertex set consisting of all k-subsets of an

n-set. Two distinct vertices are adjacent in K

G(n,k) if they have empty intersection as subs

ets.

• Since KG(n,1) = Kn , we assume k 2.

)2,5(KG

)2,5(KG 2,1

)2,5(KG 2,1

4,3

5,3 5,4

)2,5(KG 2,1

4,3

5,3 5,4

5,1

5,2

)2,5(KG 2,1

4,3

5,3 5,4

5,1

5,2

3,2

4,1 4,2

3,1

• Theorem. (Lovász； 1994)

22),( knknKG

1.

1),(* knknKG

1.

2.

1),(* knknKG

3),12(),12( * kkKGkkKG

1.

2.

3.

1),(* knknKG

3),12(),12( * kkKGkkKG

65

7

2

1)2,()2,( *

n

n

if

if

n

nnKGnKG

1.

2.

3.

4.

1),(* knknKG

3),12(),12( * kkKGkkKG

65

7

2

1)2,()2,( *

n

n

if

if

n

nnKGnKG

137

1514

16

4

3

2

)3,()3,( *

n

n

n

if

if

if

n

n

n

nKGnKG

• Sketch proof of

• S is an i-flower of KG(n,k) if any k-subset in S contains

the integer i. An i-flower is an independent set of KG(n,

k).

• It is natural to partition the flowers to form an equitable

coloring of KG(n,k). Hence, if f is an equitable m-colori

ng of KG(n,k) such that every color class under f is con

tained in some flower, then m n-k+1.

1),(* knknKG

• KG(7,2) is equitable 6-colorable.

12

13

14

15

16

17

23

24

25

26

27

34

35

36

37

45

46

47

56

57

67Y:

C(7,2)=21=4+4+4+3+3+3


12

13

14

15

16

17

23

24

25

26

27

34

35

36

37

45

46

47

56

57

67Y:

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6 6X:


12

13

14

15

16

17

23

24

25

26

27

34

35

36

37

45

46

47

56

57

67Y:

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6 6X:

…

• Theorem. (P. Hall； 1935)

A bipartite graph G = G(X,Y) with bipart

ition (X,Y) has a matching that saturates every

vertex in X if and only if |N(S)| |S| for all S

X, where N(S) denotes the set of neighbors o

f vertices in S.


12

13

14

15

16

17

23

24

25

26

27

34

35

36

37

45

46

47

56

57

67Y:

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6 6X:

V1={12,15,16,17}, V2={24,25,26,27},V3={13,23,36,37},

V4={14,34,47}, V5={35,45,57},V6={46,56,67}

• Conjecture.

for k 2. ),(),( * knKGknKG

the equitable coloring of kneser graphs 陳伯亮 ＆ 黃國卿 2008 年 8 月 11 日

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the equitable coloring of kneser graphs 陳伯亮＆黃國卿 2008 年 8 月 11 日