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 vertices have different labels. The labels are colors. The vertices with the same color form a color class.

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.

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.

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.

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. K 3,3 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.

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

K 5,8

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

Theorem 2. (Brooks, 1964) Let G be a connected graph different from K n and C 2n+1. Then

Conjecture 1. (Meyer, 1973) Let G be a connected graph different from K n and C 2n+1. Then

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

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 (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)

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 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.

Theorem 3. (Lovász, 1994)

Theorem 4. (Chen and Huang, 2008)

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 – = 6 C(7,2) = 21 =

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

Y:Y: X:X:

Y:Y: X:X: …

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 Y:Y: X: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}

The odd graph O(k) is the KG(2k+1,k). Theorem 6. (Chen and Huang, 2008)

Theorem 7. (Chen and Huang, 2008)

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.