The Equitable Colorings of Kneser Graphs

Kuo-Ching Huang (黃國卿 )

Department of Applied Mathematics

Providence University

This is a joined work with Prof. Bor-Liang Chen.

• A k-coloring of a simple graph G is a labeling f：V(G) {1,2,...,k} such that the adjacent vertic

es have different labels. The labels are colors.

The vertices with the same color form a color c

lass.

• A graph G is k-colorable if G has a k-coloring.

• The chromatic number of a graph G, denoted by

, is the least k such that G is k-colorable.)(G

• An equitable k-coloring of a graph G is a k-col

oring 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 an

equitable 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 that G

is equitably n-colorable for all n k.

),(G

),(* G

Remarks• If G is k-colorable, then G is (k +1)-coloable.

It may be happened that a graph G is equitably k-colorable, but not equitably (k +1)-coloable.

K3,3 is equitably 2-colorable, but not equitably 3-coloable.

• If H is a subgraph of G, then

It may be happened that

where H is a subgraph of G.

).()( GH ),()( GH

2)( 3,3 K 3)( 3,1 K

• By the definition, it is easy to see that

• The equalities may be not hold.

).()()( * GGG

K5,8

.5)( 8,5* K,3)( 8,5 K,2)( 8,5 K

• Theorem 1. (Hajnal and Szemerédi, 1970)

So, we have

.1)()(* GG

.1)()()()( * GGGG

• Theorem 2. (Brooks, 1964)

Let G be a connected graph different fro

m Kn and C2n+1. Then ).()( GG

• Conjecture 1. (Meyer, 1973)

Let G be a connected graph different fro

m Kn and C2n+1. Then ).()( GG

• Conjecture 2. (Chen, Lih and Wu, 1994)

A connected graph G is equitable (G)-

colorable if and only if

• Conjecture 2 implies Conjecture 1.

., 12,1212 nnnn KGandCGKG

• The Conjecture 1 is affirmative.

– Planar graphs (Yap and Zhang, 1998)

– d-degenerate graphs (Kostochka et al., 2005)

Known results

• The Conjecture 2 is affirmative.

– Graphs with Δ(G) |≧ V(G)|/2 or Δ(G) 3 (C≦hen, Lih and Wu, 1994)

– Graphs with |V(G)|/2 > Δ(G) |≧ V(G)|/3 + 1

(Yap and Zhang, 1994)

– Bipartite graphs ( Lih and Wu, 1996)

– Outplanar graphs (Yap and Zhang, 1997)

• Determine k such that G is equitable k-colorable.

– Complete r-partite graphs (Wu, 1994)

– Trees (Chen and Lih ； 1994)

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

he vertex set consisting of all k-subsets of an n-set. Two distinct vertices are adjacent in KG(n,k) if they have empty intersection as subsets.

• Since KG(n,1) = Kn , we assume k 2.

2,1

4,3

5,3 5,4

5,1

5,2

3,2

4,1 4,2

3,1

)2,5(KG

• Theorem 3. (Lovász, 1994)

.22),( knknKG

• Theorem 4. (Chen and Huang, 2008)

.1),(* knknKG

Idea of the proof of Theorem 4

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

• Any i-flower is an independent set of KG(n,k).•

• We will partition the flowers to form an equitable coloring of KG(n,k).

• If f is an equitable m-coloring of KG(n,k) such that every color class under f is contained in some flower, then m n – k +1.

KG(7,2) is equitable 6-colorable.

n – k + 1 = 7 – 2 + 1 = 6

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

KG(7,2) is equitable 6-colorable.

12

13

14

15

16

17

23

24

25

26

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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:

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:

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

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:

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

…

• Theorem 5. (P. Hall, 1935)

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

bipartition (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 of

vertices in S.

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:

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}

• The odd graph O(k) is the KG(2k+1,k).

• Theorem 6. (Chen and Huang, 2008)

.3)()( * kOkO

• Theorem 7. (Chen and Huang, 2008)

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

• Conjecture 3. (Chen and Huang, 2008)

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

• Conjecture 4.(Zhu, 2008)

For a fixed k, the equitable chromatic number of KG(n,k) is a decreasing step function with respect to n with jump one.