Chromatic Number of Grassmann Graphs and MRD codes

TL;DR

This paper establishes new bounds on the chromatic number of Grassmann graphs using MRD codes, showing asymptotic tightness in certain parameter regimes and advancing understanding of their coloring properties.

Contribution

The paper introduces a novel application of MRD codes to bound the chromatic number of Grassmann graphs, generalizing previous lifting techniques.

Findings

Derived an upper bound on the chromatic number using MRD codes.

Proved the bound is asymptotically tight for fixed parameters.

Established a new asymptotic characterization of the chromatic number.

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)}). $