Chromatic Number of Grassmann Graphs and MRD codes

TL;DR

This work studies the chromatic number of Grassmann graphs $J_q(n,m)$ and their $t$-power graphs $J_q(n,m,t)$, introducing a constructive colouring based on maximum rank distance (MRD) codes. By lifting disjoint cosets of an MRD code, the authors build cocliques tied to identifying vectors and derive upper bounds that, together with duality, extend to multiple parameter regimes; they obtain $\chi(J_q(n,m,t)) \le (1+o(1)) n^{m-t} \genfrac{[}{]}{0pt}{}{n-t}{m-t}_q$ for $n\ge 2m$ and analogous bounds for $m<n<2m$. They also provide lower bounds from clique sizes and binary/$q$-ary code considerations, and show that for fixed $(n,m,t)$, the bound is asymptotically tight, yielding a chromatic number that grows like $\Theta(q^{(m-t)\max(n-m,m)})$ in $q$ and is polynomial in $n$ with degree $m-t$. The combination of MRD-based colourings and Johnson-graph insights offers a unified asymptotic picture for the chromatic properties of Grassmann graphs and their powers, with implications for subspace coding and related combinatorial constructions.

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, Kötter and Kschischang, we use \emph{maximum rank distance (MRD)} codes to establish that $χ(J_q(n, m, t)) \leq (1 +o(1))n^{m-t}q^{(n-m)(m-t)})$ 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 $ χ(J_q(n, m, t)) = Θ(q^{(m-t)\max(n-m, m)}). $

Theorems & Definitions (19)

• Theorem 1.2: dhaeseleer2025chromatic
• Theorem 1.3
• Proposition 3.1
• proof
• Theorem 4.1
• Example 4.2
• Remark 4.3
• Lemma 4.4
• proof
• Proposition 4.5