Chromatic Number of Grassmann Graphs and MRD codes

Abstract

In this paper we investigate the chromatic number of the Grassmann graphs and of their powers, denoted $J_q(n,m,t)$. In this graph, the vertices correspond to the $m$-dimensional subspaces in $\mathbb{F}_q^n$ and two vertices are adjacent if the corresponding subspaces intersect in a subspace of dimension at least $t$. By generalizing the lifting technique of Silva, K\"otter and Kschischang, we use \emph{maximum rank distance (MRD)} codes to establish that $\chi(J_q(n, m, t)) \leq (1 +o(1))n^{m-t}q^{(n-m)(m-t)})$ when $n \geq 2m$. Given that $J_q(n, m, t)$ is isomorphic to $J_q(n,n-m,n-2m+t)$, this establishes a new upper bound on $J_q(n, m, t)$ for any valid choice of parameters. Furthermore, we observe that in the regime that $n, m $, and $t$ are fixed, our bound is asymptotically tight, implying that $ \chi(J_q(n, m, t)) = \Theta(q^{(m-t)\max(n-m, m)}). $