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 $n \geq 3k-1$. 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 $r \geq 2$. Recently, Kim and Park \cite{KP2014} showed that $\chi(K^2(2k+1,k)) \leq 2k+2$ 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 $1 \leq r \leq k -2$, (a) $\chi(K^2 (2k+r, k)) \leq (2k+r)^r$, if $2k + r = 2^t$ for some integer $t$, and (b) $\chi(K^2 (2k+r, k)) \leq (2k+r+1)^r$, if $2k + r = 2^t-1$ for some integer $t$. On the other hand, it was showed in \cite{KP2014} that $\chi(K^2 (2k+r, k)) \leq (r+2)(3k + \frac{3r+3}{2})^r$ for $2 \leq r \leq k-2$. We improve these bounds by showing that for any integer $r$ with $2 \leq r \leq k -2$, we have $\chi(K^2 (2k+r, k)) \leq2 \left(\frac{9}{4}k + \frac{9(r+3)}{8} \right)^r$. Our approach is also related with injective coloring and coloring of Johnson graph.