The Equitable Coloring of Kneser Graphs

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. K3,3

9. K5,8

10. K5,8

11. K5,8

12. K5,8

13. K5,8

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) = Kn , we assume k  2.

29. Theorem. (Lovász；1994)

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. Y: C(7,2)=21=4+4+4+3+3+3

37. KG(7,2) is equitable 6-colorable. Y: X:

38. KG(7,2) is equitable 6-colorable. Y: X:

39. KG(7,2) is equitable 6-colorable. Y: X:

40. KG(7,2) is equitable 6-colorable. Y: … 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. Y: X: 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}

43. Conjecture. for k 2.