Coloring of the Square of Kneser Graph $K(2k+r,k)$

Abstract

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 $G^2$ of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in $G^2$ 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 $n \ge 3k-1$. 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 $\chi (K^2(2k+r,k))$ for $r \ge 2$. Recently, Kim and Park (Discrete Math 315:69–74, 2014) showed that $\chi (K^2(2k+1,k)) \le 2k+2$ if $2k +1 = 2^t -1$ for some positive integer t. In this paper, we generalize the result by showing that for any integer r with $1 \le r \le k -2$,

• (a) $\chi (K^2 (2k+r, k)) \le (2k+r)^r$,   if   $2k + r = 2^t$ for some integer t, and

• (b) $\chi (K^2 (2k+r, k)) \le (2k+r+1)^r$,   if  $2k + r = 2^t-1$ for some integer t.

On the other hand, it was shown in Kim and Park (Discrete Math 315:69–74, 2014) that $\chi (K^2 (2k+r, k)) \le (r+2)(3k + \frac{3r+3}{2})^r$ for $2 \le r \le k-2$. We improve these bounds by showing that for any integer r with $2 \le r \le k -2$, we have $\chi (K^2 (2k+r, k)) \le 2 \left( \frac{9}{4}k + \frac{9(r+3)}{8} \right) ^r$. Our approach is also related with injective coloring and coloring of Johnson graph.