Coloring of the Square of Kneser Graph

The Kneser graph K (n, k) is the graph whose vertices are the k-element subsets of an n elements set, with two vertices adjacent if they 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 (n, k) is an interesting graph coloring problem, and is also related with intersecting family problem. The square of K (2k, k) is a perfect matching and the square of K (n, k) is the complete graph when n ≥ 3k − 1. Hence coloring of the square of K (2k + 1, k) has been studied as the first nontrivial case. In this paper, we focus on the question of determining χ(K 2(2k + r, k)) for r ≥ 2. Recently, Kim and Park (Discrete Math 315:69–74, 2014) showed that χ(K 2(2k + 1, k)) ≤ 2k + 2 if 2k + 1 = 2t − 1 for some positive integer t . In this paper, we generalize the result by showing that for any integer r with 1 ≤ r ≤ k − 2, (a) χ(K 2(2k + r, k)) ≤ (2k + r)r , if 2k + r = 2t for some integer t , and (b) χ(K 2(2k + r, k)) ≤ (2k + r + 1)r , if 2k + r = 2t − 1 for some integer t . On the other hand, it was shown inKim and Park (DiscreteMath 315:69–74, 2014) that χ(K 2(2k+r, k)) ≤ (r+2)(3k+ 3r+3 2 )r for 2 ≤ r ≤ k−2. We improve these bounds by showing that for any integer r with 2 ≤ r ≤ k − 2, we have χ(K 2(2k + r, k)) ≤ B Boram Park [email protected]; [email protected] Seog-Jin Kim [email protected] 1 Department of Mathematics Education, Konkuk University, Seoul, Korea 2 Department of Mathematics, Ajou University, Suwon, Korea