On the Chromatic Number of Grassmann Graphs

TL;DR

This paper investigates the chromatic number of Grassmann graphs, establishing bounds similar to Johnson graphs, and explores specific cases related to line parallelisms in projective geometries.

Contribution

It provides new bounds for the chromatic number of Grassmann graphs and connects these bounds to geometric structures like line parallelisms.

Findings

Bounds for $ ext{chi}(J_q(n,m))$ analogous to Johnson graphs

Determination of $ ext{chi}(J_q(n,2))$ relates to line parallelisms

Proved $ ext{chi}(J_q(n,2)) < 2inom{n-1}{1}_q$ for certain q and n

Abstract

In this paper we study the chromatic number of the Grassmann graphs $J_q(n, m)$. We show that $\binom{n-m+1}{1}_q \leq \chi(J_q(n, m)) \leq \binom{n}{1}_q$, which is analogous to the best-known bounds for the chromatic number of the Johnson graphs $J(n, m)$. When $m = 2$, determining $\chi(J_q(n, 2))$ 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 $\chi(J_q(n, 2))$. Finally, we prove that when $q$ is any power of two, and $n$ is any even integer, then $\chi(J_q(n, 2)) < 2\binom{n-1}{1}_q$.