The Equitable Colorings of Kneser Graphs

1. 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.

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

3. 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.

4. An equitable k-coloring of a graph G is a 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 an 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. K3,3 is equitably 2-colorable, but not equitably 3-coloable. 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.

7. If H is a subgraph of G, then It may be happened that where H is a subgraph of G.

8. By the definition, it is easy to see that • The equalities may be not hold.

9. K5,8

10. Theorem 1. (Hajnal and Szemerédi, 1970) So, we have

11. Theorem 2. (Brooks, 1964) Let G be a connected graph different from Knand C2n+1. Then

12. Conjecture 1. (Meyer, 1973) Let G be a connected graph different from Knand C2n+1. Then

13. Conjecture 2. (Chen, Lih and Wu, 1994) A connected graph G is equitable (G)-colorable if and only if • Conjecture 2 implies Conjecture 1.

14. Known results • The Conjecture 1 is affirmative. • Planar graphs (Yap and Zhang, 1998) • d-degenerate graphs (Kostochka et al., 2005)

15. The Conjecture 2 is affirmative. • Graphs with Δ(G) ≧ |V(G)|/2 or Δ(G) ≦ 3 (Chen, 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)

16. Determine k such that G is equitable k-colorable. • Complete r-partite graphs (Wu, 1994) • Trees (Chen and Lih ；1994)

17. For n 2k +1, the Kneser graphKG(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.

19. Theorem 3. (Lovász, 1994)

20. Theorem 4. (Chen and Huang, 2008)

21. 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.

23. KG(7,2) is equitable 6-colorable. n – k + 1 = 7 – 2 + 1 = 6 C(7,2) = 21 = 4 + 4 + 4 + 3 + 3 + 3

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

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

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

27. 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.

28. 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}

29. The odd graphO(k) is the KG(2k+1,k). • Theorem 6. (Chen and Huang, 2008)

30. Theorem 7. (Chen and Huang, 2008)

31. Conjecture 3. (Chen and Huang, 2008) for k 2. • 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.