Colouring lines in projective space

Let V be a vector space of dimension v over a field of order q. The q-Kneser graph has the k-dimensional subspaces of V as its vertices, where two subspaces alpha and beta are adjacent if and only if alpha boolean AND beta is the zero subspace. This paper is motivated by the problem of determining the chromatic numbers oF these graphs. This problem is trivial when k = 1 (and the graphs are complete) or when v < 2k (and the graphs are empty). We establish some basic theory in the general case. Then specializing to the case k = 2, we show that the chromatic number is q(2) + q when v = 4 and (q(v-1) - 1)/(q - 1) when v > 4. In both cases we characterise the minimal colourrings. (c) 2005 Elsevier Inc. All rights reserved.