On the Chromatic Number of Grassmann Graphs

TL;DR

This work analyzes the chromatic number $\chi(J_q(n,m))$ of Grassmann graphs, establishing a general $q$-analogue bound $\binom{n-m+1}{1}_q \le \chi(J_q(n,m)) \le \binom{n}{1}_q$ via a determinant-based coloring using cosets of $\mathbb F_{q^n}^*/\mathbb F_q^*$ and the matrix $L(x_1,\dots,x_m)$. It then specializes to the case $m=2$, connecting the coloring problem to line parallelisms in finite projective spaces and surveying known results on such parallelisms. For $q=2^e$ and even $n$, it gives a sharper bound $\chi(J_q(n,2)) < 2\binom{n-1}{1}_q$ by constructing a graph homomorphism from $J_q(n,2)$ to the 2-Kneser graph $K_2((n-1)e,e)$ and using the known chromatic number of that graph. The results bridge finite geometry and graph colorings, offering general bounds and a concrete improvement in a key parameter regime, while outlining limitations and directions for future refinements via line parallelisms and subgraph chromatic analyses.

Abstract

In this paper we study the chromatic number of the Grassmann graphs $J_q(n, m)$. We show that $\binom{n-m+1}{1}_q \leq χ(J_q(n, m)) \leq \binom{n}{1}_q$, which is analogous to the best-known bounds for the chromatic number of the Johnson graphs $J(n, m)$. When $m = 2$, determining $χ(J_q(n, 2))$ is equivalent to determining the smallest number of partial line parallelisms that one can partition the lines of PG$(n-1, q)$ into. We survey known results about line parallelisms and their implications for $χ(J_q(n, 2))$. Finally, we prove that when $q$ is any power of two, and $n$ is any even integer, then $χ(J_q(n, 2)) < 2\binom{n-1}{1}_q$.

Theorems & Definitions (13)

• Theorem 1.1: Blokhuis et. al. HM2010
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof : Proof of Theorem \ref{['MainThm1']}
• Lemma 3.1
• proof
• Lemma 3.2