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 kdimensional subspaces of V as its vertices, where two subspaces and are adjacent if and only if 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 q2 + q when v = 4 and (qv-1 - 1)/(q - 1) when v > 4. In both cases we characterise the minimal colourings.

• Type: Journal
• Year: 2006
• Where: JCT
• Authors: Ameera Chowdhury, Chris D. Godsil, Gordon F. Royle
• Keywords: Chromatic Number, JCT 2006, Kdimensional Subspaces, Vector Space