Improved bounds on the chromatic numbers of the square of Kneser graphs

The Kneser graph K(n, k) is the graph whose vertices are the k-elements subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The square G2 of a graph G is the graph defined on V (G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. Determining the chromatic number of the square of the Kneser graph K(2k + 1, k) is an interesting problem, but not much progress has been made. Kim and Nakprasit [7] showed that χ(K2(2k+1, k)) ≤ 4k+2, and Chen, Lih, and Wu [1] showed that χ(K2(2k+1, k)) ≤ 3k+2 for k ≥ 3. In this paper, we give improved upper bounds on χ(K2(2k+1, k)). We show that χ(K2(2k + 1, k)) ≤ 2k + 2, if 2k + 1 = 2n − 1 for some positive integer n. Also we show that χ(K2(2k + 1, k)) ≤ 83k + 20 3 for every integer k ≥ 2. In addition to giving improved upper bounds, our proof is concise and can be easily understood by readers while the proof in [1] is very complicated. Moreover, we show that χ(K2(2k + r, k)) = Θ(kr) for each integer 2 ≤ r ≤ k − 2.