1. The Equitable Coloring of Kneser Graphs 陳伯亮 ＆ 黃國卿 2008 年 8 月 11 日

2. A proper k-coloring of a graph G is an labeling f ： V(G)  {1,2,...,k} such that adjacent vertices have different labels. The labels are colors; The vertices of one color form a color class.

3. A graph G is k-colorable if G has a proper k- coloring. The chromatic number of a graph G, denoted by, is the least k such that G is k- colorable.

4. A equitable k-coloring of a graph G is an proper 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 equitable k-coloring.

5. 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.

6. Lemma.

7. If graph G is equitably k-colorable, then the size of all color classes in a nonincreasing sort will be or the sizes of all color classes in a nondecreasing sort will be

8. K 3,3

9. K 5,8

14. Theorem.. Theorem. (Hajnal and Szemerédi ； 1970).

15. Lemma ：

16. Theorem. (Brooks ； 1964) Let G be a connected graph. Then if

17. Conjecture. (Meyer ； 1973) Let G be a connected graph. Then if

18. Conjecture. (Chen, Lih and Wu ； 1994) A connected graph G is equitable  (G)- colorable if and only if

19. 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

20. Theorem. (Chen and Lih ； 1994) A tree T = T(X,Y), if and only if If, then

21. Theorem. (Chen and Lih ； 1994) Let T be a tree such that, then, where v is an arbitrary major vertex.

22. Theorem. (Wu ； 1994) is equitably k-colorable if and only if and for all i, where

23. 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 KG(n,k) if they have empty intersection as subsets. Since KG(n,1) = K n, we assume k  2.

29. Theorem. (Lovász ； 1994)

30. 1.

31. 1. 2.

32. 1. 2. 3.

33. 1. 2. 3. 4.

34. 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- coloring of KG(n,k) such that every color class under f is contained in some flower, then m  n-k+1.

36. KG (7,2) is equitable 6-colorable. 1212 1313 1414 1515 1616 1717 2323 2424 2525 2626 2727 3434 3535 3636 3737 4545 4646 4747 5656 5757 6767 Y: C(7,2)=21=4+4+4+3+3+3

37. KG (7,2) is equitable 6-colorable. 1212 1313 1414 1515 1616 1717 2323 2424 2525 2626 2727 3434 3535 3636 3737 4545 4646 4747 5656 5757 6767 Y: 111122223333444555666 X:

38. KG (7,2) is equitable 6-colorable. 1212 1313 1414 1515 1616 1717 2323 2424 2525 2626 2727 3434 3535 3636 3737 4545 4646 4747 5656 5757 6767 Y: 111122223333444555666 X:

39. KG (7,2) is equitable 6-colorable. 1212 1313 1414 1515 1616 1717 2323 2424 2525 2626 2727 3434 3535 3636 3737 4545 4646 4747 5656 5757 6767 Y: 111122223333444555666 X:

40. KG (7,2) is equitable 6-colorable. 1212 1313 1414 1515 1616 1717 2323 2424 2525 2626 2727 3434 3535 3636 3737 4545 4646 4747 5656 5757 6767 Y: 111122223333444555666 X: …

41. Theorem. (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.

42. KG (7,2) is equitable 6-colorable. 1212 1313 1414 1515 1616 1717 2323 2424 2525 2626 2727 3434 3535 3636 3737 4545 4646 4747 5656 5757 6767 Y: 111122223333444555666 X: V 1 ={12,15,16,17}, V 2 ={24,25,26,27},V 3 ={13,23,36,37}, V 4 ={14,34,47}, V 5 ={35,45,57},V 6 ={46,56,67}

43. Conjecture. for k  2.